| L(s) = 1 | − 12·9-s − 44·11-s + 96·23-s + 58·25-s + 68·29-s + 236·37-s + 92·43-s − 20·53-s + 44·67-s + 392·71-s + 328·79-s + 90·81-s + 528·99-s + 644·107-s − 92·109-s + 960·113-s + 570·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.09e3·169-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 4·11-s + 4.17·23-s + 2.31·25-s + 2.34·29-s + 6.37·37-s + 2.13·43-s − 0.377·53-s + 0.656·67-s + 5.52·71-s + 4.15·79-s + 10/9·81-s + 16/3·99-s + 6.01·107-s − 0.844·109-s + 8.49·113-s + 4.71·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 6.44·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(66.69919930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(66.69919930\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 58 T^{2} + 3153 T^{4} - 100898 T^{6} + 3048836 T^{8} - 100898 p^{4} T^{10} + 3153 p^{8} T^{12} - 58 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 2 p T + 441 T^{2} + 6662 T^{3} + 81668 T^{4} + 6662 p^{2} T^{5} + 441 p^{4} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 - 1090 T^{2} + 553857 T^{4} - 171023522 T^{6} + 35061921764 T^{8} - 171023522 p^{4} T^{10} + 553857 p^{8} T^{12} - 1090 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( 1 - 960 T^{2} + 606396 T^{4} - 264252480 T^{6} + 87307574150 T^{8} - 264252480 p^{4} T^{10} + 606396 p^{8} T^{12} - 960 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( 1 - 1210 T^{2} + 545409 T^{4} - 100686674 T^{6} + 10440464324 T^{8} - 100686674 p^{4} T^{10} + 545409 p^{8} T^{12} - 1210 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( ( 1 - 48 T + 1212 T^{2} - 31056 T^{3} + 873014 T^{4} - 31056 p^{2} T^{5} + 1212 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 34 T + 2301 T^{2} - 56918 T^{3} + 2679848 T^{4} - 56918 p^{2} T^{5} + 2301 p^{4} T^{6} - 34 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 4300 T^{2} + 9792666 T^{4} - 485572400 p T^{6} + 16796571875171 T^{8} - 485572400 p^{5} T^{10} + 9792666 p^{8} T^{12} - 4300 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 - 118 T + 9193 T^{2} - 502870 T^{3} + 21291460 T^{4} - 502870 p^{2} T^{5} + 9193 p^{4} T^{6} - 118 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 6168 T^{2} + 21108060 T^{4} - 52972041384 T^{6} + 102006783313478 T^{8} - 52972041384 p^{4} T^{10} + 21108060 p^{8} T^{12} - 6168 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 46 T + 3673 T^{2} - 131086 T^{3} + 8403460 T^{4} - 131086 p^{2} T^{5} + 3673 p^{4} T^{6} - 46 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 13408 T^{2} + 83515836 T^{4} - 320749243808 T^{6} + 843494155855238 T^{8} - 320749243808 p^{4} T^{10} + 83515836 p^{8} T^{12} - 13408 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 10 T + 5301 T^{2} + 123590 T^{3} + 15221312 T^{4} + 123590 p^{2} T^{5} + 5301 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 15170 T^{2} + 111771409 T^{4} - 564578054642 T^{6} + 2208988892853604 T^{8} - 564578054642 p^{4} T^{10} + 111771409 p^{8} T^{12} - 15170 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( 1 - 3560 T^{2} - 3639908 T^{4} + 13480499752 T^{6} + 27669588746374 T^{8} + 13480499752 p^{4} T^{10} - 3639908 p^{8} T^{12} - 3560 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 - 22 T + 10557 T^{2} + 5782 T^{3} + 54787352 T^{4} + 5782 p^{2} T^{5} + 10557 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 196 T + 22624 T^{2} - 2070604 T^{3} + 164062462 T^{4} - 2070604 p^{2} T^{5} + 22624 p^{4} T^{6} - 196 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 73 | \( 1 - 12690 T^{2} + 81857745 T^{4} - 215049052626 T^{6} + 493201051470692 T^{8} - 215049052626 p^{4} T^{10} + 81857745 p^{8} T^{12} - 12690 p^{12} T^{14} + p^{16} T^{16} \) |
| 79 | \( ( 1 - 164 T + 10386 T^{2} + 663008 T^{3} - 111678085 T^{4} + 663008 p^{2} T^{5} + 10386 p^{4} T^{6} - 164 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 50682 T^{2} + 1152460209 T^{4} - 15336635334306 T^{6} + 130556747398908548 T^{8} - 15336635334306 p^{4} T^{10} + 1152460209 p^{8} T^{12} - 50682 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 46416 T^{2} + 1005043164 T^{4} - 13551842433456 T^{6} + 126834279943162310 T^{8} - 13551842433456 p^{4} T^{10} + 1005043164 p^{8} T^{12} - 46416 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( 1 - 10002 T^{2} + 257682129 T^{4} - 2790925375890 T^{6} + 29923705256218532 T^{8} - 2790925375890 p^{4} T^{10} + 257682129 p^{8} T^{12} - 10002 p^{12} T^{14} + p^{16} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.35398556281148407015218347389, −3.20587498949684213356123098748, −3.17350149263021396799757983578, −3.08814479459256625497844249539, −3.05968742361012725324186435979, −2.99692692049732755172843168179, −2.76689091356676372870946961615, −2.73339219133294501585221374478, −2.65227352948941408985712362928, −2.48270353444305492115133404212, −2.37599152685096836602547832849, −2.18835831641576259669318011988, −2.09659275727117258108750693301, −2.09514167093883538097371643858, −1.90875976694140369843057526378, −1.60137213354517563939128019496, −1.52275049109963588782675296597, −0.845426980020115833767723950420, −0.827572191967142932991880668641, −0.78514715428613573000662023107, −0.77467344244955787216095697076, −0.76402220171010414617351597132, −0.75763295096597687058286177491, −0.49547582128910370093315719675, −0.35441468201695012840749056361,
0.35441468201695012840749056361, 0.49547582128910370093315719675, 0.75763295096597687058286177491, 0.76402220171010414617351597132, 0.77467344244955787216095697076, 0.78514715428613573000662023107, 0.827572191967142932991880668641, 0.845426980020115833767723950420, 1.52275049109963588782675296597, 1.60137213354517563939128019496, 1.90875976694140369843057526378, 2.09514167093883538097371643858, 2.09659275727117258108750693301, 2.18835831641576259669318011988, 2.37599152685096836602547832849, 2.48270353444305492115133404212, 2.65227352948941408985712362928, 2.73339219133294501585221374478, 2.76689091356676372870946961615, 2.99692692049732755172843168179, 3.05968742361012725324186435979, 3.08814479459256625497844249539, 3.17350149263021396799757983578, 3.20587498949684213356123098748, 3.35398556281148407015218347389
Plot not available for L-functions of degree greater than 10.
