| L(s) = 1 | − 0.325·2-s − 1.89·4-s + 3.19·5-s − 2.43·7-s + 1.26·8-s − 1.04·10-s − 11-s − 5.49·13-s + 0.794·14-s + 3.37·16-s + 2.18·17-s + 7.07·19-s − 6.05·20-s + 0.325·22-s − 6.98·23-s + 5.20·25-s + 1.78·26-s + 4.62·28-s − 1.30·29-s + 1.11·31-s − 3.63·32-s − 0.712·34-s − 7.79·35-s + 5.67·37-s − 2.30·38-s + 4.05·40-s + 8.17·41-s + ⋯ |
| L(s) = 1 | − 0.230·2-s − 0.946·4-s + 1.42·5-s − 0.922·7-s + 0.448·8-s − 0.329·10-s − 0.301·11-s − 1.52·13-s + 0.212·14-s + 0.843·16-s + 0.530·17-s + 1.62·19-s − 1.35·20-s + 0.0694·22-s − 1.45·23-s + 1.04·25-s + 0.350·26-s + 0.873·28-s − 0.242·29-s + 0.199·31-s − 0.642·32-s − 0.122·34-s − 1.31·35-s + 0.932·37-s − 0.373·38-s + 0.640·40-s + 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.289250671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.289250671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.325T + 2T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 - 2.18T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + 6.98T + 23T^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 31 | \( 1 - 1.11T + 31T^{2} \) |
| 37 | \( 1 - 5.67T + 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 + 6.15T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 + 1.99T + 53T^{2} \) |
| 59 | \( 1 + 5.77T + 59T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 3.75T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 5.85T + 89T^{2} \) |
| 97 | \( 1 - 8.17T + 97T^{2} \) |
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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
−7.958063128005340962497877626506, −7.56331989474318753937524453014, −6.55056799432739616871259170265, −5.82175277592231226883004571799, −5.29471398412896068762487450444, −4.64078328963505584422629295701, −3.53274099799182089182786710330, −2.75080086854728813659763009658, −1.83077531299893986636369950305, −0.61495544163021829447154245307,
0.61495544163021829447154245307, 1.83077531299893986636369950305, 2.75080086854728813659763009658, 3.53274099799182089182786710330, 4.64078328963505584422629295701, 5.29471398412896068762487450444, 5.82175277592231226883004571799, 6.55056799432739616871259170265, 7.56331989474318753937524453014, 7.958063128005340962497877626506
