| L(s) = 1 | + 2.18·2-s + 1.24·3-s + 2.78·4-s + 5-s + 2.71·6-s + 2.13·7-s + 1.71·8-s − 1.45·9-s + 2.18·10-s + 11-s + 3.45·12-s + 5.05·13-s + 4.67·14-s + 1.24·15-s − 1.81·16-s − 1.12·17-s − 3.18·18-s − 0.136·19-s + 2.78·20-s + 2.65·21-s + 2.18·22-s + 1.74·23-s + 2.13·24-s + 25-s + 11.0·26-s − 5.53·27-s + 5.95·28-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 0.717·3-s + 1.39·4-s + 0.447·5-s + 1.10·6-s + 0.807·7-s + 0.607·8-s − 0.485·9-s + 0.691·10-s + 0.301·11-s + 0.998·12-s + 1.40·13-s + 1.24·14-s + 0.320·15-s − 0.452·16-s − 0.273·17-s − 0.751·18-s − 0.0313·19-s + 0.622·20-s + 0.579·21-s + 0.466·22-s + 0.364·23-s + 0.435·24-s + 0.200·25-s + 2.16·26-s − 1.06·27-s + 1.12·28-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
| \(L(1)\) |
\(\approx\) |
\(7.189253306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.189253306\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.18T + 2T^{2} \) |
| 3 | \( 1 - 1.24T + 3T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 0.136T + 19T^{2} \) |
| 23 | \( 1 - 1.74T + 23T^{2} \) |
| 29 | \( 1 - 6.11T + 29T^{2} \) |
| 31 | \( 1 - 4.25T + 31T^{2} \) |
| 37 | \( 1 - 1.77T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 5.98T + 53T^{2} \) |
| 59 | \( 1 - 7.89T + 59T^{2} \) |
| 61 | \( 1 + 7.11T + 61T^{2} \) |
| 67 | \( 1 + 4.15T + 67T^{2} \) |
| 71 | \( 1 - 7.05T + 71T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 6.06T + 89T^{2} \) |
| 97 | \( 1 + 1.06T + 97T^{2} \) |
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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
−8.443076591645658971612170613837, −7.73620562895260276541018954311, −6.52131266957348647140216615819, −6.21074385235960561421281253157, −5.32474699309294406811083168833, −4.63588743958808166756045987646, −3.86607212097415204782515617019, −3.09037008168707088216416484191, −2.38669004000101613006047781036, −1.34722721862104729207894354044,
1.34722721862104729207894354044, 2.38669004000101613006047781036, 3.09037008168707088216416484191, 3.86607212097415204782515617019, 4.63588743958808166756045987646, 5.32474699309294406811083168833, 6.21074385235960561421281253157, 6.52131266957348647140216615819, 7.73620562895260276541018954311, 8.443076591645658971612170613837
