| L(s) = 1 | − 2-s − 3-s + 4-s + 3.78·5-s + 6-s − 1.64·7-s − 8-s + 9-s − 3.78·10-s − 4.73·11-s − 12-s + 5.80·13-s + 1.64·14-s − 3.78·15-s + 16-s + 4.37·17-s − 18-s − 19-s + 3.78·20-s + 1.64·21-s + 4.73·22-s − 7.00·23-s + 24-s + 9.35·25-s − 5.80·26-s − 27-s − 1.64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.69·5-s + 0.408·6-s − 0.621·7-s − 0.353·8-s + 0.333·9-s − 1.19·10-s − 1.42·11-s − 0.288·12-s + 1.61·13-s + 0.439·14-s − 0.978·15-s + 0.250·16-s + 1.06·17-s − 0.235·18-s − 0.229·19-s + 0.847·20-s + 0.358·21-s + 1.00·22-s − 1.45·23-s + 0.204·24-s + 1.87·25-s − 1.13·26-s − 0.192·27-s − 0.310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 3.78T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 5.80T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 23 | \( 1 + 7.00T + 23T^{2} \) |
| 29 | \( 1 + 6.20T + 29T^{2} \) |
| 31 | \( 1 + 7.11T + 31T^{2} \) |
| 37 | \( 1 + 0.893T + 37T^{2} \) |
| 41 | \( 1 + 4.99T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 - 0.404T + 47T^{2} \) |
| 59 | \( 1 - 5.00T + 59T^{2} \) |
| 61 | \( 1 + 0.295T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 3.04T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 6.54T + 89T^{2} \) |
| 97 | \( 1 + 2.40T + 97T^{2} \) |
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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
−7.77945622507161386285048988671, −6.90806239638933812953260409830, −6.18659783182296391716226299837, −5.56629198250804760400426569753, −5.47888524370444818193387559949, −3.92269269835449972172370601955, −3.01173145283032933517749651710, −2.03875252833173399616488369489, −1.36629981898347530261697472056, 0,
1.36629981898347530261697472056, 2.03875252833173399616488369489, 3.01173145283032933517749651710, 3.92269269835449972172370601955, 5.47888524370444818193387559949, 5.56629198250804760400426569753, 6.18659783182296391716226299837, 6.90806239638933812953260409830, 7.77945622507161386285048988671
