| L(s) = 1 | + 0.209·2-s − 1.22·3-s − 1.95·4-s − 0.258·6-s − 2.44·7-s − 0.830·8-s − 1.48·9-s − 2.39·11-s + 2.40·12-s − 0.190·13-s − 0.512·14-s + 3.73·16-s + 5.21·17-s − 0.312·18-s + 1.40·19-s + 3.00·21-s − 0.502·22-s − 5.67·23-s + 1.02·24-s − 0.0400·26-s + 5.51·27-s + 4.78·28-s + 8.18·29-s − 1.99·31-s + 2.44·32-s + 2.94·33-s + 1.09·34-s + ⋯ |
| L(s) = 1 | + 0.148·2-s − 0.710·3-s − 0.977·4-s − 0.105·6-s − 0.923·7-s − 0.293·8-s − 0.495·9-s − 0.721·11-s + 0.694·12-s − 0.0529·13-s − 0.137·14-s + 0.934·16-s + 1.26·17-s − 0.0735·18-s + 0.322·19-s + 0.656·21-s − 0.107·22-s − 1.18·23-s + 0.208·24-s − 0.00785·26-s + 1.06·27-s + 0.903·28-s + 1.51·29-s − 0.358·31-s + 0.432·32-s + 0.512·33-s + 0.187·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 - 0.209T + 2T^{2} \) |
| 3 | \( 1 + 1.22T + 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 + 0.190T + 13T^{2} \) |
| 17 | \( 1 - 5.21T + 17T^{2} \) |
| 19 | \( 1 - 1.40T + 19T^{2} \) |
| 23 | \( 1 + 5.67T + 23T^{2} \) |
| 29 | \( 1 - 8.18T + 29T^{2} \) |
| 31 | \( 1 + 1.99T + 31T^{2} \) |
| 37 | \( 1 - 7.49T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 3.60T + 43T^{2} \) |
| 47 | \( 1 - 0.599T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 1.88T + 67T^{2} \) |
| 71 | \( 1 + 3.97T + 71T^{2} \) |
| 73 | \( 1 - 3.17T + 73T^{2} \) |
| 79 | \( 1 + 7.95T + 79T^{2} \) |
| 83 | \( 1 + 8.39T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 - 8.97T + 97T^{2} \) |
| 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
−8.136318057991017884120372876681, −7.08262591622937181668435882072, −6.21581855049938004139597044590, −5.60930030857261300108653238946, −5.16155892505088371188084739291, −4.21113375255676152658357744908, −3.36265629060283159304357206564, −2.65531691259051274139482461074, −0.941982740346138751930070943118, 0,
0.941982740346138751930070943118, 2.65531691259051274139482461074, 3.36265629060283159304357206564, 4.21113375255676152658357744908, 5.16155892505088371188084739291, 5.60930030857261300108653238946, 6.21581855049938004139597044590, 7.08262591622937181668435882072, 8.136318057991017884120372876681
