| L(s) = 1 | + 1.19·3-s − 1.76·5-s + 4.20·7-s − 1.57·9-s + 6.20·11-s + 6.78·13-s − 2.11·15-s − 4.29·17-s + 2.98·19-s + 5.02·21-s + 4.21·23-s − 1.87·25-s − 5.46·27-s + 0.933·29-s + 4.70·31-s + 7.40·33-s − 7.43·35-s − 11.8·37-s + 8.09·39-s + 7.43·41-s + 7.15·43-s + 2.78·45-s − 11.7·47-s + 10.6·49-s − 5.13·51-s + 4.02·53-s − 10.9·55-s + ⋯ |
| L(s) = 1 | + 0.689·3-s − 0.790·5-s + 1.58·7-s − 0.524·9-s + 1.86·11-s + 1.88·13-s − 0.544·15-s − 1.04·17-s + 0.684·19-s + 1.09·21-s + 0.878·23-s − 0.375·25-s − 1.05·27-s + 0.173·29-s + 0.844·31-s + 1.28·33-s − 1.25·35-s − 1.94·37-s + 1.29·39-s + 1.16·41-s + 1.09·43-s + 0.414·45-s − 1.70·47-s + 1.52·49-s − 0.718·51-s + 0.552·53-s − 1.47·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.064414224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.064414224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 7 | \( 1 - 4.20T + 7T^{2} \) |
| 11 | \( 1 - 6.20T + 11T^{2} \) |
| 13 | \( 1 - 6.78T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 2.98T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 - 0.933T + 29T^{2} \) |
| 31 | \( 1 - 4.70T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 7.43T + 41T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 4.02T + 53T^{2} \) |
| 59 | \( 1 + 9.10T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 - 3.31T + 71T^{2} \) |
| 73 | \( 1 + 6.72T + 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 6.83T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.593421209428342299588891267486, −7.962518560543930735241520221632, −7.08091358968983013628345876868, −6.33467472319977717444403843037, −5.43334218225435278074148421640, −4.36339207905413495486378246084, −3.91736989177573915002616914304, −3.12874089153698072058436243801, −1.80436330526890416089463946996, −1.09262056208496493735861799200,
1.09262056208496493735861799200, 1.80436330526890416089463946996, 3.12874089153698072058436243801, 3.91736989177573915002616914304, 4.36339207905413495486378246084, 5.43334218225435278074148421640, 6.33467472319977717444403843037, 7.08091358968983013628345876868, 7.962518560543930735241520221632, 8.593421209428342299588891267486
