| L(s) = 1 | − 0.287·2-s − 3-s − 1.91·4-s + 5-s + 0.287·6-s − 1.07·7-s + 1.12·8-s + 9-s − 0.287·10-s − 2.40·11-s + 1.91·12-s + 13-s + 0.309·14-s − 15-s + 3.51·16-s + 0.378·17-s − 0.287·18-s + 5.04·19-s − 1.91·20-s + 1.07·21-s + 0.692·22-s + 3.38·23-s − 1.12·24-s + 25-s − 0.287·26-s − 27-s + 2.06·28-s + ⋯ |
| L(s) = 1 | − 0.203·2-s − 0.577·3-s − 0.958·4-s + 0.447·5-s + 0.117·6-s − 0.406·7-s + 0.398·8-s + 0.333·9-s − 0.0908·10-s − 0.726·11-s + 0.553·12-s + 0.277·13-s + 0.0826·14-s − 0.258·15-s + 0.877·16-s + 0.0917·17-s − 0.0677·18-s + 1.15·19-s − 0.428·20-s + 0.234·21-s + 0.147·22-s + 0.705·23-s − 0.229·24-s + 0.200·25-s − 0.0563·26-s − 0.192·27-s + 0.390·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9782834080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9782834080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.287T + 2T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 + 2.40T + 11T^{2} \) |
| 17 | \( 1 - 0.378T + 17T^{2} \) |
| 19 | \( 1 - 5.04T + 19T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 + 3.29T + 29T^{2} \) |
| 37 | \( 1 - 8.21T + 37T^{2} \) |
| 41 | \( 1 + 0.318T + 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 + 5.16T + 47T^{2} \) |
| 53 | \( 1 + 9.13T + 53T^{2} \) |
| 59 | \( 1 + 4.64T + 59T^{2} \) |
| 61 | \( 1 - 2.40T + 61T^{2} \) |
| 67 | \( 1 + 8.33T + 67T^{2} \) |
| 71 | \( 1 + 5.08T + 71T^{2} \) |
| 73 | \( 1 + 0.510T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 9.57T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 9.26T + 97T^{2} \) |
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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
−7.929212136685626951747536953559, −7.57673895607267248249060651767, −6.51088648347070159862776701566, −5.89585165162893310404884862998, −5.11555396085607219683585613817, −4.72971534957013146712128536140, −3.62780155867395356370114304626, −2.91079508475794177967444283350, −1.56080237868884140496604179085, −0.58370590263550753297257535961,
0.58370590263550753297257535961, 1.56080237868884140496604179085, 2.91079508475794177967444283350, 3.62780155867395356370114304626, 4.72971534957013146712128536140, 5.11555396085607219683585613817, 5.89585165162893310404884862998, 6.51088648347070159862776701566, 7.57673895607267248249060651767, 7.929212136685626951747536953559
