| L(s) = 1 | − 2-s − 3-s + 4-s + 3.28·5-s + 6-s − 3.26·7-s − 8-s + 9-s − 3.28·10-s + 3.43·11-s − 12-s + 13-s + 3.26·14-s − 3.28·15-s + 16-s − 4.30·17-s − 18-s − 4.26·19-s + 3.28·20-s + 3.26·21-s − 3.43·22-s + 0.898·23-s + 24-s + 5.77·25-s − 26-s − 27-s − 3.26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.46·5-s + 0.408·6-s − 1.23·7-s − 0.353·8-s + 0.333·9-s − 1.03·10-s + 1.03·11-s − 0.288·12-s + 0.277·13-s + 0.872·14-s − 0.847·15-s + 0.250·16-s − 1.04·17-s − 0.235·18-s − 0.978·19-s + 0.734·20-s + 0.712·21-s − 0.733·22-s + 0.187·23-s + 0.204·24-s + 1.15·25-s − 0.196·26-s − 0.192·27-s − 0.616·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 3.28T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 + 4.26T + 19T^{2} \) |
| 23 | \( 1 - 0.898T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 - 0.0760T + 31T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 + 0.879T + 53T^{2} \) |
| 59 | \( 1 + 8.39T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 - 2.20T + 67T^{2} \) |
| 71 | \( 1 + 9.35T + 71T^{2} \) |
| 73 | \( 1 - 7.76T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 5.25T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 0.220T + 97T^{2} \) |
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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
−7.16109945348469433572727576896, −6.65199529672841081348770502190, −6.22812241260508876691007839195, −5.81879042782935666461936595502, −4.75908600795292261878333231330, −3.87094831256749968418056838684, −2.85440007280551662543352117777, −2.03852573706414966735884581159, −1.22779395800480222303085605842, 0,
1.22779395800480222303085605842, 2.03852573706414966735884581159, 2.85440007280551662543352117777, 3.87094831256749968418056838684, 4.75908600795292261878333231330, 5.81879042782935666461936595502, 6.22812241260508876691007839195, 6.65199529672841081348770502190, 7.16109945348469433572727576896
