| L(s) = 1 | − 3-s + 1.68·5-s − 0.722·7-s + 9-s − 4.92·11-s − 5.15·13-s − 1.68·15-s + 4.50·17-s − 3.23·19-s + 0.722·21-s − 9.11·23-s − 2.14·25-s − 27-s + 1.58·29-s + 2.13·31-s + 4.92·33-s − 1.21·35-s − 8.13·37-s + 5.15·39-s + 5.37·41-s + 4.74·43-s + 1.68·45-s + 3.74·47-s − 6.47·49-s − 4.50·51-s + 1.47·53-s − 8.31·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·5-s − 0.272·7-s + 0.333·9-s − 1.48·11-s − 1.43·13-s − 0.435·15-s + 1.09·17-s − 0.741·19-s + 0.157·21-s − 1.90·23-s − 0.429·25-s − 0.192·27-s + 0.293·29-s + 0.383·31-s + 0.857·33-s − 0.206·35-s − 1.33·37-s + 0.825·39-s + 0.839·41-s + 0.723·43-s + 0.251·45-s + 0.546·47-s − 0.925·49-s − 0.630·51-s + 0.202·53-s − 1.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9286599749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9286599749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 + 0.722T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 + 9.11T + 23T^{2} \) |
| 29 | \( 1 - 1.58T + 29T^{2} \) |
| 31 | \( 1 - 2.13T + 31T^{2} \) |
| 37 | \( 1 + 8.13T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 - 4.74T + 43T^{2} \) |
| 47 | \( 1 - 3.74T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 2.51T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 2.96T + 83T^{2} \) |
| 89 | \( 1 - 7.54T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−7.77131251551152383827071859036, −7.19998001919254685737594910527, −6.23569789031746598853315762511, −5.76664121632894636375667781244, −5.16285548015989648338853417034, −4.50621473444740034649585240633, −3.47980234319214260784952624541, −2.43106412704242453815552797767, −1.98420257226293536166453664776, −0.46108285108728015078980882500,
0.46108285108728015078980882500, 1.98420257226293536166453664776, 2.43106412704242453815552797767, 3.47980234319214260784952624541, 4.50621473444740034649585240633, 5.16285548015989648338853417034, 5.76664121632894636375667781244, 6.23569789031746598853315762511, 7.19998001919254685737594910527, 7.77131251551152383827071859036
