| L(s) = 1 | − 0.679·3-s − 1.36·5-s + 2.99·7-s − 2.53·9-s + 4.20·11-s + 1.86·13-s + 0.926·15-s + 0.227·17-s − 3.54·19-s − 2.03·21-s + 2.07·23-s − 3.14·25-s + 3.76·27-s − 4.12·29-s − 2.59·31-s − 2.85·33-s − 4.07·35-s − 4.32·37-s − 1.27·39-s − 6.83·41-s − 3.66·43-s + 3.46·45-s + 7.50·47-s + 1.94·49-s − 0.154·51-s + 7.77·53-s − 5.73·55-s + ⋯ |
| L(s) = 1 | − 0.392·3-s − 0.609·5-s + 1.13·7-s − 0.846·9-s + 1.26·11-s + 0.518·13-s + 0.239·15-s + 0.0552·17-s − 0.812·19-s − 0.443·21-s + 0.431·23-s − 0.628·25-s + 0.724·27-s − 0.765·29-s − 0.466·31-s − 0.497·33-s − 0.689·35-s − 0.710·37-s − 0.203·39-s − 1.06·41-s − 0.558·43-s + 0.515·45-s + 1.09·47-s + 0.278·49-s − 0.0216·51-s + 1.06·53-s − 0.773·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 0.679T + 3T^{2} \) |
| 5 | \( 1 + 1.36T + 5T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 - 0.227T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 - 2.07T + 23T^{2} \) |
| 29 | \( 1 + 4.12T + 29T^{2} \) |
| 31 | \( 1 + 2.59T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 + 6.83T + 41T^{2} \) |
| 43 | \( 1 + 3.66T + 43T^{2} \) |
| 47 | \( 1 - 7.50T + 47T^{2} \) |
| 53 | \( 1 - 7.77T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 + 4.86T + 61T^{2} \) |
| 67 | \( 1 - 7.83T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 2.78T + 79T^{2} \) |
| 83 | \( 1 + 3.68T + 83T^{2} \) |
| 89 | \( 1 + 0.715T + 89T^{2} \) |
| 97 | \( 1 + 0.611T + 97T^{2} \) |
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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
−7.42636244053016357030028332330, −6.86504867363723789811308359785, −5.98971250392043909293239216374, −5.48236318978639368116954642986, −4.59629132007430315365481048976, −3.98314333799459263356664092704, −3.27871050981518340840112677220, −2.04940374756777276635509352639, −1.26119536406320076054571516030, 0,
1.26119536406320076054571516030, 2.04940374756777276635509352639, 3.27871050981518340840112677220, 3.98314333799459263356664092704, 4.59629132007430315365481048976, 5.48236318978639368116954642986, 5.98971250392043909293239216374, 6.86504867363723789811308359785, 7.42636244053016357030028332330
