| L(s) = 1 | − 3-s + 1.38·5-s − 3.43·7-s + 9-s + 2.64·11-s + 3.17·13-s − 1.38·15-s − 7.44·17-s + 7.20·19-s + 3.43·21-s + 23-s − 3.07·25-s − 27-s + 29-s − 6.70·31-s − 2.64·33-s − 4.76·35-s + 2.03·37-s − 3.17·39-s + 7.14·41-s − 3.08·43-s + 1.38·45-s + 5.04·47-s + 4.80·49-s + 7.44·51-s + 9.24·53-s + 3.66·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.619·5-s − 1.29·7-s + 0.333·9-s + 0.798·11-s + 0.881·13-s − 0.357·15-s − 1.80·17-s + 1.65·19-s + 0.749·21-s + 0.208·23-s − 0.615·25-s − 0.192·27-s + 0.185·29-s − 1.20·31-s − 0.460·33-s − 0.804·35-s + 0.334·37-s − 0.509·39-s + 1.11·41-s − 0.470·43-s + 0.206·45-s + 0.736·47-s + 0.686·49-s + 1.04·51-s + 1.27·53-s + 0.494·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.519055093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.519055093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 + 3.43T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 + 7.44T + 17T^{2} \) |
| 19 | \( 1 - 7.20T + 19T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 2.03T + 37T^{2} \) |
| 41 | \( 1 - 7.14T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 - 5.04T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 - 0.888T + 59T^{2} \) |
| 61 | \( 1 + 0.116T + 61T^{2} \) |
| 67 | \( 1 + 7.88T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 8.68T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 8.50T + 97T^{2} \) |
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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
−7.61055490384054859556912306253, −6.94259986219880366367031241934, −6.35797485903441638608344471063, −5.93071996588580335368383774840, −5.22556438623135775601690725932, −4.17212616081814113361132371130, −3.62088154794836981244158807587, −2.68185551038203717761494357737, −1.67758382359400430339514539404, −0.63752559154714647095328542268,
0.63752559154714647095328542268, 1.67758382359400430339514539404, 2.68185551038203717761494357737, 3.62088154794836981244158807587, 4.17212616081814113361132371130, 5.22556438623135775601690725932, 5.93071996588580335368383774840, 6.35797485903441638608344471063, 6.94259986219880366367031241934, 7.61055490384054859556912306253
