| L(s) = 1 | + 1.97·2-s − 1.85·3-s + 2.90·4-s + 0.676·5-s − 3.66·6-s − 0.529·7-s + 3.76·8-s + 2.44·9-s + 1.33·10-s − 1.08·11-s − 5.39·12-s − 1.04·14-s − 1.25·15-s + 4.54·16-s + 4.82·18-s + 1.96·20-s + 0.983·21-s − 2.14·22-s − 6.99·24-s − 0.542·25-s − 2.67·27-s − 1.54·28-s + 0.955·29-s − 2.47·30-s + 5.20·32-s + 2.01·33-s − 0.358·35-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 1.85·3-s + 2.90·4-s + 0.676·5-s − 3.66·6-s − 0.529·7-s + 3.76·8-s + 2.44·9-s + 1.33·10-s − 1.08·11-s − 5.39·12-s − 1.04·14-s − 1.25·15-s + 4.54·16-s + 4.82·18-s + 1.96·20-s + 0.983·21-s − 2.14·22-s − 6.99·24-s − 0.542·25-s − 2.67·27-s − 1.54·28-s + 0.955·29-s − 2.47·30-s + 5.20·32-s + 2.01·33-s − 0.358·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.125270043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.125270043\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1151 | \( 1 - T \) |
| good | 2 | \( 1 - 1.97T + T^{2} \) |
| 3 | \( 1 + 1.85T + T^{2} \) |
| 5 | \( 1 - 0.676T + T^{2} \) |
| 7 | \( 1 + 0.529T + T^{2} \) |
| 11 | \( 1 + 1.08T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.955T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.99T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.79T + T^{2} \) |
| 47 | \( 1 - 0.380T + T^{2} \) |
| 53 | \( 1 + 1.94T + T^{2} \) |
| 59 | \( 1 + 1.54T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.229T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.71T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.54456549659335005330948024456, −9.811390179525892974838901460316, −7.71024119476570508841215050758, −6.88373942777204772102932645812, −6.19791461641944570396461320755, −5.67592438931900317751975725529, −5.03814101178392768022942430188, −4.30735549583313803623206140378, −3.03161116066479649969759704375, −1.72878795897856266977179637549,
1.72878795897856266977179637549, 3.03161116066479649969759704375, 4.30735549583313803623206140378, 5.03814101178392768022942430188, 5.67592438931900317751975725529, 6.19791461641944570396461320755, 6.88373942777204772102932645812, 7.71024119476570508841215050758, 9.811390179525892974838901460316, 10.54456549659335005330948024456
