| L(s) = 1 | + 2.16·2-s + 3.16·3-s + 2.69·4-s + 5-s + 6.85·6-s + 0.605·7-s + 1.51·8-s + 7.01·9-s + 2.16·10-s − 11-s + 8.53·12-s + 4.28·13-s + 1.31·14-s + 3.16·15-s − 2.11·16-s + 3.11·17-s + 15.2·18-s − 6.25·19-s + 2.69·20-s + 1.91·21-s − 2.16·22-s − 6.48·23-s + 4.78·24-s + 25-s + 9.27·26-s + 12.7·27-s + 1.63·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.82·3-s + 1.34·4-s + 0.447·5-s + 2.80·6-s + 0.228·7-s + 0.534·8-s + 2.33·9-s + 0.685·10-s − 0.301·11-s + 2.46·12-s + 1.18·13-s + 0.350·14-s + 0.817·15-s − 0.529·16-s + 0.755·17-s + 3.58·18-s − 1.43·19-s + 0.603·20-s + 0.418·21-s − 0.462·22-s − 1.35·23-s + 0.977·24-s + 0.200·25-s + 1.81·26-s + 2.44·27-s + 0.308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.720398060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.720398060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 3 | \( 1 - 3.16T + 3T^{2} \) |
| 7 | \( 1 - 0.605T + 7T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 + 6.25T + 19T^{2} \) |
| 23 | \( 1 + 6.48T + 23T^{2} \) |
| 29 | \( 1 - 9.98T + 29T^{2} \) |
| 31 | \( 1 + 4.22T + 31T^{2} \) |
| 37 | \( 1 + 0.294T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 4.52T + 43T^{2} \) |
| 47 | \( 1 + 7.92T + 47T^{2} \) |
| 53 | \( 1 + 9.50T + 53T^{2} \) |
| 59 | \( 1 - 0.855T + 59T^{2} \) |
| 61 | \( 1 + 0.0428T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 79 | \( 1 + 7.56T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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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
−8.266520128389746804404278899513, −7.966197458149035758240043437467, −6.58308410353012908679950386388, −6.42363193629974208629290310411, −5.18939916637015647957488775169, −4.50136869350509667896010415645, −3.61400166524641172351849423014, −3.29321218724834220559961166602, −2.25921650202664648003145207387, −1.68303397746403448140986047995,
1.68303397746403448140986047995, 2.25921650202664648003145207387, 3.29321218724834220559961166602, 3.61400166524641172351849423014, 4.50136869350509667896010415645, 5.18939916637015647957488775169, 6.42363193629974208629290310411, 6.58308410353012908679950386388, 7.966197458149035758240043437467, 8.266520128389746804404278899513
