| L(s) = 1 | + 0.430·2-s − 3-s − 1.81·4-s + 3.94·5-s − 0.430·6-s − 0.533·7-s − 1.64·8-s + 9-s + 1.69·10-s + 1.81·12-s − 4.86·13-s − 0.229·14-s − 3.94·15-s + 2.92·16-s − 0.271·17-s + 0.430·18-s + 19-s − 7.16·20-s + 0.533·21-s − 8.91·23-s + 1.64·24-s + 10.5·25-s − 2.09·26-s − 27-s + 0.969·28-s + 7.17·29-s − 1.69·30-s + ⋯ |
| L(s) = 1 | + 0.304·2-s − 0.577·3-s − 0.907·4-s + 1.76·5-s − 0.175·6-s − 0.201·7-s − 0.580·8-s + 0.333·9-s + 0.537·10-s + 0.523·12-s − 1.35·13-s − 0.0613·14-s − 1.01·15-s + 0.731·16-s − 0.0658·17-s + 0.101·18-s + 0.229·19-s − 1.60·20-s + 0.116·21-s − 1.85·23-s + 0.334·24-s + 2.11·25-s − 0.410·26-s − 0.192·27-s + 0.183·28-s + 1.33·29-s − 0.310·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.430T + 2T^{2} \) |
| 5 | \( 1 - 3.94T + 5T^{2} \) |
| 7 | \( 1 + 0.533T + 7T^{2} \) |
| 13 | \( 1 + 4.86T + 13T^{2} \) |
| 17 | \( 1 + 0.271T + 17T^{2} \) |
| 23 | \( 1 + 8.91T + 23T^{2} \) |
| 29 | \( 1 - 7.17T + 29T^{2} \) |
| 31 | \( 1 - 0.707T + 31T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 + 6.33T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 7.39T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 4.43T + 67T^{2} \) |
| 71 | \( 1 + 7.72T + 71T^{2} \) |
| 73 | \( 1 - 3.98T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 8.44T + 83T^{2} \) |
| 89 | \( 1 - 3.51T + 89T^{2} \) |
| 97 | \( 1 + 3.85T + 97T^{2} \) |
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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
−7.60204568346563192762872359683, −6.53701656984189329435408675231, −6.06740744994719176616227866554, −5.55125751037032647813657106925, −4.78006402816583459719367848480, −4.36042913011493078061022325435, −3.05338570084854205276662617576, −2.30936239766838494177400140994, −1.28582710166846434804722157050, 0,
1.28582710166846434804722157050, 2.30936239766838494177400140994, 3.05338570084854205276662617576, 4.36042913011493078061022325435, 4.78006402816583459719367848480, 5.55125751037032647813657106925, 6.06740744994719176616227866554, 6.53701656984189329435408675231, 7.60204568346563192762872359683
