| L(s) = 1 | + 1.16·3-s − 9.90·5-s − 23.9·7-s − 25.6·9-s − 28.2·11-s − 56.2·13-s − 11.5·15-s + 44.1·17-s + 19·19-s − 27.9·21-s − 134.·23-s − 26.9·25-s − 61.3·27-s − 54.2·29-s − 181.·31-s − 32.9·33-s + 237.·35-s − 4.65·37-s − 65.5·39-s − 326.·41-s + 109.·43-s + 253.·45-s + 86.8·47-s + 231.·49-s + 51.4·51-s + 208.·53-s + 279.·55-s + ⋯ |
| L(s) = 1 | + 0.224·3-s − 0.885·5-s − 1.29·7-s − 0.949·9-s − 0.773·11-s − 1.20·13-s − 0.198·15-s + 0.629·17-s + 0.229·19-s − 0.290·21-s − 1.22·23-s − 0.215·25-s − 0.437·27-s − 0.347·29-s − 1.05·31-s − 0.173·33-s + 1.14·35-s − 0.0206·37-s − 0.269·39-s − 1.24·41-s + 0.389·43-s + 0.841·45-s + 0.269·47-s + 0.675·49-s + 0.141·51-s + 0.540·53-s + 0.685·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2006323304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2006323304\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 1.16T + 27T^{2} \) |
| 5 | \( 1 + 9.90T + 125T^{2} \) |
| 7 | \( 1 + 23.9T + 343T^{2} \) |
| 11 | \( 1 + 28.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.1T + 4.91e3T^{2} \) |
| 23 | \( 1 + 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 54.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 4.65T + 5.06e4T^{2} \) |
| 41 | \( 1 + 326.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 109.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 86.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 208.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 45.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 37.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 576.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 663.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 620.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 468.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 905.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 216.T + 9.12e5T^{2} \) |
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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
−9.487818434228173395915728330575, −8.426677441821788691645788525078, −7.73136902702904501427935404330, −7.06643759661667604176530540177, −5.93493890740601580892455272328, −5.21386505731309580970318955892, −3.89367844016949114732764617606, −3.19397948564209565975714462287, −2.29648510495669281948961572439, −0.20744729023882552803903774969,
0.20744729023882552803903774969, 2.29648510495669281948961572439, 3.19397948564209565975714462287, 3.89367844016949114732764617606, 5.21386505731309580970318955892, 5.93493890740601580892455272328, 7.06643759661667604176530540177, 7.73136902702904501427935404330, 8.426677441821788691645788525078, 9.487818434228173395915728330575
