| L(s) = 1 | + 3-s − 2·4-s − 5-s − 5·7-s + 9-s + 11-s − 2·12-s + 13-s − 15-s + 4·16-s + 5·17-s − 6·19-s + 2·20-s − 5·21-s + 3·23-s + 25-s + 27-s + 10·28-s − 31-s + 33-s + 5·35-s − 2·36-s + 37-s + 39-s + 7·41-s − 2·43-s − 2·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 1.88·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 0.277·13-s − 0.258·15-s + 16-s + 1.21·17-s − 1.37·19-s + 0.447·20-s − 1.09·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s + 1.88·28-s − 0.179·31-s + 0.174·33-s + 0.845·35-s − 1/3·36-s + 0.164·37-s + 0.160·39-s + 1.09·41-s − 0.304·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−7.73004092908555084514846207333, −7.13125620748997434586158251001, −6.23635287164785815447628333953, −5.72247935647873116788107307119, −4.57046971991680212292218858288, −3.87732892475053050315358976421, −3.38208910431746833595730370283, −2.64507937441203372392430038639, −1.07616629361938923619504944306, 0,
1.07616629361938923619504944306, 2.64507937441203372392430038639, 3.38208910431746833595730370283, 3.87732892475053050315358976421, 4.57046971991680212292218858288, 5.72247935647873116788107307119, 6.23635287164785815447628333953, 7.13125620748997434586158251001, 7.73004092908555084514846207333
