| L(s) = 1 | + 8·3-s + 250·5-s + 1.28e3·7-s − 4.14e3·9-s − 2.94e3·11-s + 2.02e3·13-s + 2.00e3·15-s − 2.76e4·17-s − 5.34e4·19-s + 1.03e4·21-s − 4.21e4·23-s + 4.68e4·25-s − 4.92e4·27-s + 1.43e5·29-s + 2.55e5·31-s − 2.35e4·33-s + 3.22e5·35-s − 3.24e5·37-s + 1.62e4·39-s − 2.57e5·41-s − 5.94e5·43-s − 1.03e6·45-s − 2.16e6·47-s + 6.32e5·49-s − 2.21e5·51-s + 8.49e5·53-s − 7.36e5·55-s + ⋯ |
| L(s) = 1 | + 0.171·3-s + 0.894·5-s + 1.41·7-s − 1.89·9-s − 0.666·11-s + 0.256·13-s + 0.153·15-s − 1.36·17-s − 1.78·19-s + 0.242·21-s − 0.721·23-s + 3/5·25-s − 0.481·27-s + 1.09·29-s + 1.54·31-s − 0.114·33-s + 1.26·35-s − 1.05·37-s + 0.0437·39-s − 0.582·41-s − 1.13·43-s − 1.69·45-s − 3.03·47-s + 0.767·49-s − 0.233·51-s + 0.783·53-s − 0.596·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 8 T + 1402 p T^{2} - 8 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 184 p T + 1026822 T^{2} - 184 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2944 T + 23912102 T^{2} + 2944 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 12 p^{2} T - 18566866 T^{2} - 12 p^{9} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 27644 T + 1003140326 T^{2} + 27644 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 53488 T + 2419893558 T^{2} + 53488 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 42120 T + 1988603110 T^{2} + 42120 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 143228 T + 23837125550 T^{2} - 143228 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 255664 T + 49689334302 T^{2} - 255664 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 324164 T + 44114242206 T^{2} + 324164 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 257180 T + 239116486262 T^{2} + 257180 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 594296 T + 547062034782 T^{2} + 594296 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2160920 T + 2179210856630 T^{2} + 2160920 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 849244 T + 2527962442814 T^{2} - 849244 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2244720 T + 5847312305638 T^{2} + 2244720 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 116332 T + 1428444411342 T^{2} - 116332 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4929384 T + 17911553086574 T^{2} + 4929384 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3456432 T + 4024210501582 T^{2} + 3456432 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1393940 T + 18822156961494 T^{2} - 1393940 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11340512 T + 63037703406558 T^{2} + 11340512 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7882552 T + 67314163208654 T^{2} + 7882552 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6735180 T + 93023747263222 T^{2} + 6735180 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 23786180 T + 282313784137926 T^{2} - 23786180 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14772006264373876722981098798, −10.99969548857378377480819266226, −10.21903723766387343973849389642, −10.08849643893949614490462982162, −8.954811735414327210833822610267, −8.662761439707074937527711038283, −8.214286869160025791170042453404, −8.158537527774950879707737730724, −6.97936128394522212983004594740, −6.30516343367657591791843328163, −6.07818187733516293934560771361, −5.27753150677662411931272104999, −4.74655147458613468629570661384, −4.35261514222622243195959360605, −3.05662067078714248750721564340, −2.69200116998520241427590857406, −1.89378544176497622987284752018, −1.54784254967133588191367978551, 0, 0,
1.54784254967133588191367978551, 1.89378544176497622987284752018, 2.69200116998520241427590857406, 3.05662067078714248750721564340, 4.35261514222622243195959360605, 4.74655147458613468629570661384, 5.27753150677662411931272104999, 6.07818187733516293934560771361, 6.30516343367657591791843328163, 6.97936128394522212983004594740, 8.158537527774950879707737730724, 8.214286869160025791170042453404, 8.662761439707074937527711038283, 8.954811735414327210833822610267, 10.08849643893949614490462982162, 10.21903723766387343973849389642, 10.99969548857378377480819266226, 11.14772006264373876722981098798
