| L(s) = 1 | + 0.481·3-s − 5-s − 4.15·7-s − 2.76·9-s − 3.19·11-s + 6.63·13-s − 0.481·15-s + 2·17-s + 19-s − 2·21-s + 4.15·23-s + 25-s − 2.77·27-s + 7.73·29-s + 4.57·31-s − 1.53·33-s + 4.15·35-s + 4.63·37-s + 3.19·39-s − 1.73·41-s − 12.3·43-s + 2.76·45-s − 4.15·47-s + 10.2·49-s + 0.962·51-s − 0.637·53-s + 3.19·55-s + ⋯ |
| L(s) = 1 | + 0.277·3-s − 0.447·5-s − 1.57·7-s − 0.922·9-s − 0.963·11-s + 1.84·13-s − 0.124·15-s + 0.485·17-s + 0.229·19-s − 0.436·21-s + 0.866·23-s + 0.200·25-s − 0.534·27-s + 1.43·29-s + 0.821·31-s − 0.267·33-s + 0.702·35-s + 0.762·37-s + 0.511·39-s − 0.271·41-s − 1.88·43-s + 0.412·45-s − 0.606·47-s + 1.46·49-s + 0.134·51-s − 0.0875·53-s + 0.430·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 0.481T + 3T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 11 | \( 1 + 3.19T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 37 | \( 1 - 4.63T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 4.15T + 47T^{2} \) |
| 53 | \( 1 + 0.637T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 - 4.57T + 71T^{2} \) |
| 73 | \( 1 - 0.261T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 - 1.10T + 97T^{2} \) |
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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
−8.048220248727163283787391360377, −6.81270428850960777155687697103, −6.38031310062026990225985139029, −5.70515092866181007260563194121, −4.85811350275164414858076884842, −3.74462723324323709367970730428, −3.10759920936491044041559212853, −2.79190976770020417797210846418, −1.14165833108343166890997115044, 0,
1.14165833108343166890997115044, 2.79190976770020417797210846418, 3.10759920936491044041559212853, 3.74462723324323709367970730428, 4.85811350275164414858076884842, 5.70515092866181007260563194121, 6.38031310062026990225985139029, 6.81270428850960777155687697103, 8.048220248727163283787391360377
