| L(s) = 1 | + 32·7-s − 162·9-s + 1.84e3·17-s + 7.16e3·23-s + 9.74e3·25-s + 7.07e3·31-s + 5.03e3·41-s + 9.15e3·47-s − 1.77e4·49-s − 5.18e3·63-s + 1.51e5·71-s + 3.42e4·73-s + 2.00e5·79-s + 1.96e4·81-s − 1.80e5·89-s − 1.99e5·97-s + 6.79e5·103-s − 4.98e4·113-s + 5.91e4·119-s + 5.53e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2.99e5·153-s + ⋯ |
| L(s) = 1 | + 0.246·7-s − 2/3·9-s + 1.55·17-s + 2.82·23-s + 3.11·25-s + 1.32·31-s + 0.467·41-s + 0.604·47-s − 1.05·49-s − 0.164·63-s + 3.55·71-s + 0.751·73-s + 3.62·79-s + 1/3·81-s − 2.41·89-s − 2.15·97-s + 6.30·103-s − 0.367·113-s + 0.382·119-s + 3.43·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 1.03·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(14.50471917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.50471917\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 9748 T^{2} + 41725526 T^{4} - 9748 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 16 T + 9278 T^{2} - 16 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 553804 T^{2} + 126701631830 T^{4} - 553804 p^{10} T^{6} + p^{20} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1079252 T^{2} + 557988097398 T^{4} - 1079252 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 924 T + 414054 T^{2} - 924 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8230380 T^{2} + 28977086949238 T^{4} - 8230380 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 3584 T + 15611566 T^{2} - 3584 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 36903156 T^{2} + 892166002897910 T^{4} - 36903156 p^{10} T^{6} + p^{20} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 3536 T + 38130350 T^{2} - 3536 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 220032564 T^{2} + 21466526966405398 T^{4} - 220032564 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2516 T + 67624822 T^{2} - 2516 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 77492172 T^{2} + 41298446982584278 T^{4} - 77492172 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4576 T + 377252254 T^{2} - 4576 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1670187732 T^{2} + 1047155709832775318 T^{4} - 1670187732 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1321495820 T^{2} + 1013409615111355158 T^{4} - 1321495820 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2092476436 T^{2} + 2164810071205318326 T^{4} - 2092476436 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2098688428 T^{2} + 4067732954700328758 T^{4} - 2098688428 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 75520 T + 4788320398 T^{2} - 75520 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 17100 T + 1464583286 T^{2} - 17100 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 100496 T + 8128509326 T^{2} - 100496 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 3511348972 T^{2} + 9858377737524874358 T^{4} - 3511348972 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 90356 T + 13201612438 T^{2} + 90356 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 99908 T + 11886506054 T^{2} + 99908 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.22389235893441805097457433885, −6.90556246976098208076317117744, −6.80918636141872275608186113331, −6.71226403776852344796752957725, −6.69831513012711029328083015543, −5.83719906415459862032236820079, −5.80916242068781421686423237694, −5.69902717528854870006121370445, −5.24197762689927679558157646159, −4.87730550419806592209847963965, −4.77749477918835798638942269796, −4.71545321484724398245756813679, −4.40984184116916670436371883376, −3.57221204918745890128651046505, −3.56550592592813708777298798513, −3.22472596883268952607725979834, −3.14918572776035498027851076618, −2.61334028888696257925956979323, −2.51413706224288725525413787398, −2.07170483953381837350562980818, −1.59725889170531427467972966817, −1.00841589622890036598179101750, −0.947581064079578506487763700778, −0.70896206413391415431876176214, −0.51887285860952504813816193945,
0.51887285860952504813816193945, 0.70896206413391415431876176214, 0.947581064079578506487763700778, 1.00841589622890036598179101750, 1.59725889170531427467972966817, 2.07170483953381837350562980818, 2.51413706224288725525413787398, 2.61334028888696257925956979323, 3.14918572776035498027851076618, 3.22472596883268952607725979834, 3.56550592592813708777298798513, 3.57221204918745890128651046505, 4.40984184116916670436371883376, 4.71545321484724398245756813679, 4.77749477918835798638942269796, 4.87730550419806592209847963965, 5.24197762689927679558157646159, 5.69902717528854870006121370445, 5.80916242068781421686423237694, 5.83719906415459862032236820079, 6.69831513012711029328083015543, 6.71226403776852344796752957725, 6.80918636141872275608186113331, 6.90556246976098208076317117744, 7.22389235893441805097457433885
