| L(s) = 1 | + 3·2-s − 5·4-s − 25·5-s + 14·7-s − 33·8-s − 75·10-s + 22·11-s + 77·13-s + 42·14-s − 21·16-s − 74·17-s − 101·19-s + 125·20-s + 66·22-s − 58·23-s + 257·25-s + 231·26-s − 70·28-s − 91·29-s + 152·31-s + 87·32-s − 222·34-s − 350·35-s + 619·37-s − 303·38-s + 825·40-s − 138·41-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 5/8·4-s − 2.23·5-s + 0.755·7-s − 1.45·8-s − 2.37·10-s + 0.603·11-s + 1.64·13-s + 0.801·14-s − 0.328·16-s − 1.05·17-s − 1.21·19-s + 1.39·20-s + 0.639·22-s − 0.525·23-s + 2.05·25-s + 1.74·26-s − 0.472·28-s − 0.582·29-s + 0.880·31-s + 0.480·32-s − 1.11·34-s − 1.69·35-s + 2.75·37-s − 1.29·38-s + 3.26·40-s − 0.525·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_4$ | \( 1 - 3 T + 7 p T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p^{2} T + 368 T^{2} + p^{5} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 77 T + 5872 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 74 T + 11178 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 101 T + 15550 T^{2} + 101 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 58 T - 3402 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 91 T - 3420 T^{2} + 91 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 152 T + 52030 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 619 T + 184240 T^{2} - 619 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 138 T - 10822 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 230 T + 159846 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 149 T - 9678 T^{2} + 149 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1228 T + 12726 p T^{2} + 1228 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 11 p T + 397530 T^{2} + 11 p^{4} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 412 T + 283150 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1243 T + 975850 T^{2} + 1243 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 960 T + 684830 T^{2} + 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 741 T + 911220 T^{2} + 741 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 492 T + 759294 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1744 T + 1705670 T^{2} - 1744 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1552 T + 1996814 T^{2} - 1552 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 440 T + 521838 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.695100650929052484064759932399, −9.291499249620192903756288632822, −8.800852156284164029161057675573, −8.541905274809256627457802939734, −7.975928920412199641014663447205, −7.911556582000346082722831442695, −7.37463526735258782032998359880, −6.51316556420642225018441894384, −6.06906923320637088527070728540, −6.02916609716696278955398001203, −4.76354185403175943781663675783, −4.64842969665414758982860636284, −4.19144808374299801080415889668, −4.15265534440161621478869135058, −3.28789387465603811234845267913, −3.19288720211848933869566176116, −1.89128208761620914925301564810, −1.14519559924680600583646109908, 0, 0,
1.14519559924680600583646109908, 1.89128208761620914925301564810, 3.19288720211848933869566176116, 3.28789387465603811234845267913, 4.15265534440161621478869135058, 4.19144808374299801080415889668, 4.64842969665414758982860636284, 4.76354185403175943781663675783, 6.02916609716696278955398001203, 6.06906923320637088527070728540, 6.51316556420642225018441894384, 7.37463526735258782032998359880, 7.911556582000346082722831442695, 7.975928920412199641014663447205, 8.541905274809256627457802939734, 8.800852156284164029161057675573, 9.291499249620192903756288632822, 9.695100650929052484064759932399
