| L(s) = 1 | + 4·3-s + 2·4-s + 14·9-s + 8·12-s + 24·13-s + 16-s + 24·17-s + 12·23-s − 16·25-s + 40·27-s − 48·29-s + 28·36-s + 96·39-s + 64·43-s + 4·48-s + 96·51-s + 48·52-s + 12·61-s − 2·64-s + 48·68-s + 48·69-s − 64·75-s + 4·79-s + 99·81-s − 192·87-s + 24·92-s − 32·100-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 4-s + 14/3·9-s + 2.30·12-s + 6.65·13-s + 1/4·16-s + 5.82·17-s + 2.50·23-s − 3.19·25-s + 7.69·27-s − 8.91·29-s + 14/3·36-s + 15.3·39-s + 9.75·43-s + 0.577·48-s + 13.4·51-s + 6.65·52-s + 1.53·61-s − 1/4·64-s + 5.82·68-s + 5.77·69-s − 7.39·75-s + 0.450·79-s + 11·81-s − 20.5·87-s + 2.50·92-s − 3.19·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(162.5526043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(162.5526043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | \( 1 \) |
| 13 | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 6 T^{2} - 13 p T^{4} - 378 T^{6} + 13044 T^{8} - 378 p^{2} T^{10} - 13 p^{5} T^{12} + 6 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 + 8 T^{2} + 478 T^{4} - 9088 T^{6} + 17971 T^{8} - 9088 p^{2} T^{10} + 478 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 6 T - 17 T^{2} - 42 T^{3} + 1452 T^{4} - 42 p T^{5} - 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( 1 + 86 T^{2} + 3697 T^{4} + 152822 T^{6} + 5651524 T^{8} + 152822 p^{2} T^{10} + 3697 p^{4} T^{12} + 86 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 + 66 T^{2} + 1681 T^{4} - 4158 T^{6} - 681900 T^{8} - 4158 p^{2} T^{10} + 1681 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 16 T + 132 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 + 166 T^{2} + 16321 T^{4} + 1131622 T^{6} + 60474340 T^{8} + 1131622 p^{2} T^{10} + 16321 p^{4} T^{12} + 166 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 104 T^{2} + 8007 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 128 T^{2} + 7918 T^{4} + 192512 T^{6} + 2523091 T^{8} + 192512 p^{2} T^{10} + 7918 p^{4} T^{12} + 128 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 54 T^{2} - 3263 T^{4} - 151146 T^{6} + 4938996 T^{8} - 151146 p^{2} T^{10} - 3263 p^{4} T^{12} + 54 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 120 T^{2} + 9074 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 98 T^{2} + 3745 T^{4} - 470302 T^{6} - 45250076 T^{8} - 470302 p^{2} T^{10} + 3745 p^{4} T^{12} + 98 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 2 T + 7 T^{2} + 322 T^{3} - 6548 T^{4} + 322 p T^{5} + 7 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 256 T^{2} + 29874 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 50 T^{2} - 5421 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 242 T^{2} + 33171 T^{4} - 242 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.93173231952684096624524649308, −3.89452531040063567596275718942, −3.72123741545090016368429909591, −3.69001318138893961245448015768, −3.67450252589541431697189939056, −3.64266175876248658669802552466, −3.44453449994589605098146518534, −3.38631186975210108207248249787, −3.07505658163140637495022317396, −3.07345714665724409133409360635, −2.71604529606233176801313580720, −2.71328610496357834666117734227, −2.59621046696196246266253291575, −2.28362669221746494553466677042, −2.11611840769720655637247115400, −2.10550422174978301402244359158, −1.80579657584580413874904327044, −1.75518812845097959904020311810, −1.31210497300662148809503578252, −1.30560393672778075701108290535, −1.27439380469876428388181039276, −1.25294547642499291354910718544, −1.13112473351896484230478135045, −0.69354644684877042286805953488, −0.64748811921795507724060268162,
0.64748811921795507724060268162, 0.69354644684877042286805953488, 1.13112473351896484230478135045, 1.25294547642499291354910718544, 1.27439380469876428388181039276, 1.30560393672778075701108290535, 1.31210497300662148809503578252, 1.75518812845097959904020311810, 1.80579657584580413874904327044, 2.10550422174978301402244359158, 2.11611840769720655637247115400, 2.28362669221746494553466677042, 2.59621046696196246266253291575, 2.71328610496357834666117734227, 2.71604529606233176801313580720, 3.07345714665724409133409360635, 3.07505658163140637495022317396, 3.38631186975210108207248249787, 3.44453449994589605098146518534, 3.64266175876248658669802552466, 3.67450252589541431697189939056, 3.69001318138893961245448015768, 3.72123741545090016368429909591, 3.89452531040063567596275718942, 3.93173231952684096624524649308
Plot not available for L-functions of degree greater than 10.
