L(s) = 1 | + 0.899·2-s − 1.19·4-s − 1.67·5-s − 1.75·7-s − 2.87·8-s − 1.50·10-s + 5.08·11-s + 6.43·13-s − 1.58·14-s − 0.201·16-s + 2.24·17-s − 3.96·19-s + 1.99·20-s + 4.57·22-s − 7.71·23-s − 2.19·25-s + 5.79·26-s + 2.09·28-s − 0.101·29-s − 4.63·31-s + 5.55·32-s + 2.01·34-s + 2.94·35-s − 7.78·37-s − 3.56·38-s + 4.80·40-s − 1.42·41-s + ⋯ |
L(s) = 1 | + 0.636·2-s − 0.595·4-s − 0.748·5-s − 0.664·7-s − 1.01·8-s − 0.476·10-s + 1.53·11-s + 1.78·13-s − 0.422·14-s − 0.0504·16-s + 0.543·17-s − 0.908·19-s + 0.445·20-s + 0.976·22-s − 1.60·23-s − 0.439·25-s + 1.13·26-s + 0.395·28-s − 0.0188·29-s − 0.832·31-s + 0.982·32-s + 0.345·34-s + 0.497·35-s − 1.27·37-s − 0.578·38-s + 0.759·40-s − 0.222·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
good | 2 | \( 1 - 0.899T + 2T^{2} \) |
| 5 | \( 1 + 1.67T + 5T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 11 | \( 1 - 5.08T + 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 + 3.96T + 19T^{2} \) |
| 23 | \( 1 + 7.71T + 23T^{2} \) |
| 29 | \( 1 + 0.101T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 + 3.65T + 43T^{2} \) |
| 47 | \( 1 - 7.73T + 47T^{2} \) |
| 53 | \( 1 + 1.61T + 53T^{2} \) |
| 59 | \( 1 - 2.46T + 59T^{2} \) |
| 61 | \( 1 + 4.77T + 61T^{2} \) |
| 67 | \( 1 + 3.42T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 0.423T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 6.06T + 97T^{2} \) |
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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
−8.652509837680412428708847280399, −8.145961070990418909055293572800, −6.90074473282054767965109109121, −6.16849316257898540005045346355, −5.65526709536943049566243859939, −4.19500101962515673629936425440, −3.90097865914353830232270829547, −3.30760451202736388203106044519, −1.52783842239567128542582705179, 0,
1.52783842239567128542582705179, 3.30760451202736388203106044519, 3.90097865914353830232270829547, 4.19500101962515673629936425440, 5.65526709536943049566243859939, 6.16849316257898540005045346355, 6.90074473282054767965109109121, 8.145961070990418909055293572800, 8.652509837680412428708847280399