| L(s) = 1 | + 2-s − 0.894·3-s + 4-s − 0.0669·5-s − 0.894·6-s + 1.42·7-s + 8-s − 2.19·9-s − 0.0669·10-s − 3.23·11-s − 0.894·12-s + 4.17·13-s + 1.42·14-s + 0.0599·15-s + 16-s − 0.463·17-s − 2.19·18-s + 19-s − 0.0669·20-s − 1.27·21-s − 3.23·22-s − 2.96·23-s − 0.894·24-s − 4.99·25-s + 4.17·26-s + 4.65·27-s + 1.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.516·3-s + 0.5·4-s − 0.0299·5-s − 0.365·6-s + 0.540·7-s + 0.353·8-s − 0.733·9-s − 0.0211·10-s − 0.974·11-s − 0.258·12-s + 1.15·13-s + 0.382·14-s + 0.0154·15-s + 0.250·16-s − 0.112·17-s − 0.518·18-s + 0.229·19-s − 0.0149·20-s − 0.279·21-s − 0.689·22-s − 0.617·23-s − 0.182·24-s − 0.999·25-s + 0.819·26-s + 0.895·27-s + 0.270·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 0.894T + 3T^{2} \) |
| 5 | \( 1 + 0.0669T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 - 4.17T + 13T^{2} \) |
| 17 | \( 1 + 0.463T + 17T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 + 5.02T + 31T^{2} \) |
| 37 | \( 1 + 5.53T + 37T^{2} \) |
| 41 | \( 1 - 3.26T + 41T^{2} \) |
| 43 | \( 1 - 0.640T + 43T^{2} \) |
| 47 | \( 1 - 0.940T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 3.02T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 + 2.88T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 1.25T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.45972333187088991726008034914, −6.55122532375464468048464325790, −5.93682704262252002199688813798, −5.42940609949994448286765565631, −4.81369658380621443657557288955, −3.95799066707751006092068112338, −3.19242887913744645865452132067, −2.34256483258567364599128122881, −1.36859176921252077902887965310, 0,
1.36859176921252077902887965310, 2.34256483258567364599128122881, 3.19242887913744645865452132067, 3.95799066707751006092068112338, 4.81369658380621443657557288955, 5.42940609949994448286765565631, 5.93682704262252002199688813798, 6.55122532375464468048464325790, 7.45972333187088991726008034914
