| L(s) = 1 | − 2.36·3-s + 5-s − 3.61·7-s + 2.61·9-s − 2.20·11-s + 2.57·13-s − 2.36·15-s + 0.922·17-s − 19-s + 8.55·21-s + 7.23·23-s + 25-s + 0.922·27-s + 3.81·29-s − 4.48·31-s + 5.22·33-s − 3.61·35-s − 4.13·37-s − 6.09·39-s − 4.73·41-s + 5.42·43-s + 2.61·45-s + 0.279·47-s + 6.03·49-s − 2.18·51-s + 9.86·53-s − 2.20·55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 0.447·5-s − 1.36·7-s + 0.870·9-s − 0.664·11-s + 0.713·13-s − 0.611·15-s + 0.223·17-s − 0.229·19-s + 1.86·21-s + 1.50·23-s + 0.200·25-s + 0.177·27-s + 0.708·29-s − 0.805·31-s + 0.908·33-s − 0.610·35-s − 0.679·37-s − 0.975·39-s − 0.739·41-s + 0.827·43-s + 0.389·45-s + 0.0407·47-s + 0.862·49-s − 0.306·51-s + 1.35·53-s − 0.297·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.36T + 3T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 17 | \( 1 - 0.922T + 17T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 + 4.48T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 - 5.42T + 43T^{2} \) |
| 47 | \( 1 - 0.279T + 47T^{2} \) |
| 53 | \( 1 - 9.86T + 53T^{2} \) |
| 59 | \( 1 - 1.33T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 9.28T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 + 9.42T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 2.75T + 97T^{2} \) |
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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
−8.489931935634196828590174659428, −7.18290129491978550211666720231, −6.73998813722149471656576712966, −5.94853921636936552470648549119, −5.51459978193983725857057882649, −4.67696534222110823777317785651, −3.51554257892508421128949565773, −2.67794593956357104884326778219, −1.14927739946792770729571113009, 0,
1.14927739946792770729571113009, 2.67794593956357104884326778219, 3.51554257892508421128949565773, 4.67696534222110823777317785651, 5.51459978193983725857057882649, 5.94853921636936552470648549119, 6.73998813722149471656576712966, 7.18290129491978550211666720231, 8.489931935634196828590174659428
