L(s) = 1 | − 2-s + 2.11·3-s + 4-s − 3.59·5-s − 2.11·6-s − 7-s − 8-s + 1.45·9-s + 3.59·10-s + 3.24·11-s + 2.11·12-s + 0.613·13-s + 14-s − 7.58·15-s + 16-s − 4.69·17-s − 1.45·18-s − 3.59·20-s − 2.11·21-s − 3.24·22-s + 4.53·23-s − 2.11·24-s + 7.89·25-s − 0.613·26-s − 3.25·27-s − 28-s + 4.45·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.21·3-s + 0.5·4-s − 1.60·5-s − 0.862·6-s − 0.377·7-s − 0.353·8-s + 0.486·9-s + 1.13·10-s + 0.977·11-s + 0.609·12-s + 0.170·13-s + 0.267·14-s − 1.95·15-s + 0.250·16-s − 1.13·17-s − 0.343·18-s − 0.802·20-s − 0.460·21-s − 0.691·22-s + 0.944·23-s − 0.431·24-s + 1.57·25-s − 0.120·26-s − 0.626·27-s − 0.188·28-s + 0.826·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 2.11T + 3T^{2} \) |
| 5 | \( 1 + 3.59T + 5T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 - 0.613T + 13T^{2} \) |
| 17 | \( 1 + 4.69T + 17T^{2} \) |
| 23 | \( 1 - 4.53T + 23T^{2} \) |
| 29 | \( 1 - 4.45T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 - 5.34T + 37T^{2} \) |
| 41 | \( 1 + 1.08T + 41T^{2} \) |
| 43 | \( 1 - 5.44T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 6.12T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 6.41T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 6.43T + 79T^{2} \) |
| 83 | \( 1 + 8.52T + 83T^{2} \) |
| 89 | \( 1 - 1.49T + 89T^{2} \) |
| 97 | \( 1 - 1.67T + 97T^{2} \) |
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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
−7.943804175276135854389858633263, −7.37466306218266279955618555298, −6.85924983936196814105581671346, −5.95801872946194430510069494409, −4.50393998828091163128787758424, −3.96954147807527950421810919550, −3.20542517492370352127111250131, −2.55611680922922566712144181291, −1.28010251928211353701233256848, 0,
1.28010251928211353701233256848, 2.55611680922922566712144181291, 3.20542517492370352127111250131, 3.96954147807527950421810919550, 4.50393998828091163128787758424, 5.95801872946194430510069494409, 6.85924983936196814105581671346, 7.37466306218266279955618555298, 7.943804175276135854389858633263