| L(s) = 1 | + 2.44·3-s + 3.86·5-s − 7-s + 2.99·9-s − 2.82·11-s − 6.69·13-s + 9.46·15-s + 3.46·17-s − 2.44·19-s − 2.44·21-s + 9.46·23-s + 9.92·25-s + 3.48·29-s + 7.46·31-s − 6.92·33-s − 3.86·35-s + 0.656·37-s − 16.3·39-s + 3.46·41-s + 10.5·43-s + 11.5·45-s + 11.4·47-s + 49-s + 8.48·51-s − 1.41·53-s − 10.9·55-s − 5.99·57-s + ⋯ |
| L(s) = 1 | + 1.41·3-s + 1.72·5-s − 0.377·7-s + 0.999·9-s − 0.852·11-s − 1.85·13-s + 2.44·15-s + 0.840·17-s − 0.561·19-s − 0.534·21-s + 1.97·23-s + 1.98·25-s + 0.647·29-s + 1.34·31-s − 1.20·33-s − 0.653·35-s + 0.107·37-s − 2.62·39-s + 0.541·41-s + 1.60·43-s + 1.72·45-s + 1.67·47-s + 0.142·49-s + 1.18·51-s − 0.194·53-s − 1.47·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.509738440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.509738440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 6.69T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 - 9.46T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 37 | \( 1 - 0.656T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 1.41T + 53T^{2} \) |
| 59 | \( 1 + 3.20T + 59T^{2} \) |
| 61 | \( 1 + 7.45T + 61T^{2} \) |
| 67 | \( 1 - 4.14T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 3.20T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 + 2.39T + 97T^{2} \) |
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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
−7.88114566898001329255772902994, −7.36713480445370460099411406489, −6.63623509187006500798887881020, −5.76475903395308503178234715575, −5.11853810559936119591495172813, −4.42396777479809998010694104705, −3.03285118110679281472323427357, −2.64605676839955958583977646105, −2.26001329646534628181926116532, −1.02026170067251437294194629744,
1.02026170067251437294194629744, 2.26001329646534628181926116532, 2.64605676839955958583977646105, 3.03285118110679281472323427357, 4.42396777479809998010694104705, 5.11853810559936119591495172813, 5.76475903395308503178234715575, 6.63623509187006500798887881020, 7.36713480445370460099411406489, 7.88114566898001329255772902994
