L(s) = 1 | + (0.995 + 0.0980i)2-s + (0.980 + 0.195i)4-s + (0.290 + 0.956i)7-s + (0.956 + 0.290i)8-s + (0.881 − 0.471i)9-s + (−1.46 + 0.217i)11-s + (0.195 + 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−1.48 + 0.0727i)22-s + (−0.938 + 0.0924i)23-s + (0.634 + 0.773i)25-s + (0.0980 + 0.995i)28-s + (−0.509 − 0.686i)29-s + (0.881 + 0.471i)32-s + ⋯ |
L(s) = 1 | + (0.995 + 0.0980i)2-s + (0.980 + 0.195i)4-s + (0.290 + 0.956i)7-s + (0.956 + 0.290i)8-s + (0.881 − 0.471i)9-s + (−1.46 + 0.217i)11-s + (0.195 + 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−1.48 + 0.0727i)22-s + (−0.938 + 0.0924i)23-s + (0.634 + 0.773i)25-s + (0.0980 + 0.995i)28-s + (−0.509 − 0.686i)29-s + (0.881 + 0.471i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.187088680\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.187088680\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.995 - 0.0980i)T \) |
| 7 | \( 1 + (-0.290 - 0.956i)T \) |
good | 3 | \( 1 + (-0.881 + 0.471i)T^{2} \) |
| 5 | \( 1 + (-0.634 - 0.773i)T^{2} \) |
| 11 | \( 1 + (1.46 - 0.217i)T + (0.956 - 0.290i)T^{2} \) |
| 13 | \( 1 + (0.773 + 0.634i)T^{2} \) |
| 17 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 19 | \( 1 + (-0.0980 - 0.995i)T^{2} \) |
| 23 | \( 1 + (0.938 - 0.0924i)T + (0.980 - 0.195i)T^{2} \) |
| 29 | \( 1 + (0.509 + 0.686i)T + (-0.290 + 0.956i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.0970 + 1.97i)T + (-0.995 - 0.0980i)T^{2} \) |
| 41 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 43 | \( 1 + (-0.390 + 1.55i)T + (-0.881 - 0.471i)T^{2} \) |
| 47 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 53 | \( 1 + (1.02 - 1.37i)T + (-0.290 - 0.956i)T^{2} \) |
| 59 | \( 1 + (0.773 - 0.634i)T^{2} \) |
| 61 | \( 1 + (0.471 + 0.881i)T^{2} \) |
| 67 | \( 1 + (-0.416 - 0.249i)T + (0.471 + 0.881i)T^{2} \) |
| 71 | \( 1 + (0.523 - 0.979i)T + (-0.555 - 0.831i)T^{2} \) |
| 73 | \( 1 + (0.831 + 0.555i)T^{2} \) |
| 79 | \( 1 + (0.858 + 1.28i)T + (-0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (0.995 - 0.0980i)T^{2} \) |
| 89 | \( 1 + (-0.980 - 0.195i)T^{2} \) |
| 97 | \( 1 + (0.707 + 0.707i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.612721295211697890659456314552, −8.653730779342128524102701232597, −7.63041160637274278444218294948, −7.25412594105632122883921865312, −6.02044089391216490193638083623, −5.52329030748289410489117978255, −4.66947594398018657142118767155, −3.78204940657488801280691559955, −2.66487111335675428671348558560, −1.86292524611070978520708540665,
1.44886703288893351813861400783, 2.60422658405687382780388609780, 3.61242318929886344217790303223, 4.66909501747735729606668604357, 4.97247672573988628862864906290, 6.17430708375190221979768074760, 6.95226224550278214818243665432, 7.78743368627955490545643058229, 8.172500851504084808599844860128, 9.875008587743448087470272522953