L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−1.19 + 1.59i)5-s + (−0.415 + 0.909i)8-s + (−0.281 − 0.959i)9-s + (1.41 + 1.41i)10-s + (−0.340 − 1.56i)13-s + (0.841 + 0.540i)16-s + (0.797 − 1.74i)17-s + (−0.989 + 0.142i)18-s + (1.59 − 1.19i)20-s + (−0.839 − 2.85i)25-s + (−1.59 + 0.114i)26-s + (−1.03 + 0.304i)29-s + (0.654 − 0.755i)32-s + ⋯ |
L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−1.19 + 1.59i)5-s + (−0.415 + 0.909i)8-s + (−0.281 − 0.959i)9-s + (1.41 + 1.41i)10-s + (−0.340 − 1.56i)13-s + (0.841 + 0.540i)16-s + (0.797 − 1.74i)17-s + (−0.989 + 0.142i)18-s + (1.59 − 1.19i)20-s + (−0.839 − 2.85i)25-s + (−1.59 + 0.114i)26-s + (−1.03 + 0.304i)29-s + (0.654 − 0.755i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5575931795\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5575931795\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 353 | \( 1 + (-0.415 - 0.909i)T \) |
good | 3 | \( 1 + (0.281 + 0.959i)T^{2} \) |
| 5 | \( 1 + (1.19 - 1.59i)T + (-0.281 - 0.959i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 13 | \( 1 + (0.340 + 1.56i)T + (-0.909 + 0.415i)T^{2} \) |
| 17 | \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 23 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 29 | \( 1 + (1.03 - 0.304i)T + (0.841 - 0.540i)T^{2} \) |
| 31 | \( 1 + (0.989 + 0.142i)T^{2} \) |
| 37 | \( 1 + (0.244 + 0.654i)T + (-0.755 + 0.654i)T^{2} \) |
| 41 | \( 1 + (0.627 - 0.544i)T + (0.142 - 0.989i)T^{2} \) |
| 43 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 53 | \( 1 + (-1.40 - 1.05i)T + (0.281 + 0.959i)T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (0.909 + 0.584i)T + (0.415 + 0.909i)T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + (-0.540 + 0.841i)T^{2} \) |
| 73 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 83 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 89 | \( 1 + (-0.697 + 1.86i)T + (-0.755 - 0.654i)T^{2} \) |
| 97 | \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.748921008007013496365046780659, −8.724583259122681516642517952420, −7.69948978207250586108044315464, −7.22693148756424972339493224871, −6.04331637635835761438433403597, −5.09561501003587730634231941921, −3.80396411010210754589966700838, −3.20999237337213880063082540578, −2.66844643626319750903032149966, −0.45713985828832129268359005991,
1.58588897563417691520473191964, 3.74063680518018310908696559038, 4.27980784303002222711922315659, 5.07743807936713642844401765192, 5.78308005289293992618444632019, 7.01534791381241297972783181562, 7.82730553298251313445042836983, 8.314457439189834374993893118971, 8.921010862782902114100269194567, 9.695948636540775230498809561171