| L(s) = 1 | + 9.57·3-s − 2.39·5-s + 7·7-s + 64.5·9-s + 21.2·11-s − 31.4·13-s − 22.8·15-s + 38.5·17-s − 94.2·19-s + 66.9·21-s + 128.·23-s − 119.·25-s + 359.·27-s + 291.·29-s + 64.5·31-s + 203.·33-s − 16.7·35-s + 157.·37-s − 300.·39-s − 41·41-s + 335.·43-s − 154.·45-s + 404.·47-s + 49·49-s + 369.·51-s − 304.·53-s − 50.8·55-s + ⋯ |
| L(s) = 1 | + 1.84·3-s − 0.213·5-s + 0.377·7-s + 2.39·9-s + 0.582·11-s − 0.670·13-s − 0.393·15-s + 0.550·17-s − 1.13·19-s + 0.696·21-s + 1.16·23-s − 0.954·25-s + 2.56·27-s + 1.86·29-s + 0.374·31-s + 1.07·33-s − 0.0808·35-s + 0.698·37-s − 1.23·39-s − 0.156·41-s + 1.18·43-s − 0.511·45-s + 1.25·47-s + 0.142·49-s + 1.01·51-s − 0.788·53-s − 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.772203797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.772203797\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 - 9.57T + 27T^{2} \) |
| 5 | \( 1 + 2.39T + 125T^{2} \) |
| 11 | \( 1 - 21.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 157.T + 5.06e4T^{2} \) |
| 43 | \( 1 - 335.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 404.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 304.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 149.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 785.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 582.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 310.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 676.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 954.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.221433176693582739318064814856, −8.629773226166349430406110030000, −7.88209169209718111881149142086, −7.29003154875896958771801728766, −6.29325251909298578322103734404, −4.71520020726187053889199003254, −4.08898365616505295279521668675, −3.03076003386589089643007662881, −2.27440809373341692087755227199, −1.11842747692810932539920455773,
1.11842747692810932539920455773, 2.27440809373341692087755227199, 3.03076003386589089643007662881, 4.08898365616505295279521668675, 4.71520020726187053889199003254, 6.29325251909298578322103734404, 7.29003154875896958771801728766, 7.88209169209718111881149142086, 8.629773226166349430406110030000, 9.221433176693582739318064814856
