| L(s) = 1 | + 4.47·17-s − 4·19-s − 8.94·23-s − 8·31-s + 8.94·47-s − 7·49-s − 4.47·53-s + 2·61-s − 16·79-s + 17.8·83-s − 17.8·107-s − 14·109-s − 4.47·113-s + ⋯ |
| L(s) = 1 | + 1.08·17-s − 0.917·19-s − 1.86·23-s − 1.43·31-s + 1.30·47-s − 49-s − 0.614·53-s + 0.256·61-s − 1.80·79-s + 1.96·83-s − 1.72·107-s − 1.34·109-s − 0.420·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 17.8T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.059492953441077715721486581214, −7.61201764492633572828264296859, −6.64110706226956238394306696091, −5.91829988567615582973787674166, −5.26066871631210117691861364451, −4.19477066569243289082313729622, −3.58919177503697992681102310607, −2.45602772843344046098083229385, −1.51742371842981158249503352735, 0,
1.51742371842981158249503352735, 2.45602772843344046098083229385, 3.58919177503697992681102310607, 4.19477066569243289082313729622, 5.26066871631210117691861364451, 5.91829988567615582973787674166, 6.64110706226956238394306696091, 7.61201764492633572828264296859, 8.059492953441077715721486581214
