| L(s) = 1 | + 2.58·2-s + 1.69·3-s + 4.68·4-s − 5-s + 4.38·6-s + 2.28·7-s + 6.95·8-s − 0.129·9-s − 2.58·10-s + 11-s + 7.94·12-s − 1.76·13-s + 5.91·14-s − 1.69·15-s + 8.60·16-s − 7.54·17-s − 0.333·18-s − 19-s − 4.68·20-s + 3.87·21-s + 2.58·22-s − 6.89·23-s + 11.7·24-s + 25-s − 4.56·26-s − 5.30·27-s + 10.7·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 0.978·3-s + 2.34·4-s − 0.447·5-s + 1.78·6-s + 0.864·7-s + 2.45·8-s − 0.0430·9-s − 0.817·10-s + 0.301·11-s + 2.29·12-s − 0.489·13-s + 1.58·14-s − 0.437·15-s + 2.15·16-s − 1.83·17-s − 0.0787·18-s − 0.229·19-s − 1.04·20-s + 0.846·21-s + 0.551·22-s − 1.43·23-s + 2.40·24-s + 0.200·25-s − 0.894·26-s − 1.02·27-s + 2.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.053010874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.053010874\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 7.54T + 17T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 - 8.39T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 4.07T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 8.70T + 73T^{2} \) |
| 79 | \( 1 - 4.40T + 79T^{2} \) |
| 83 | \( 1 - 0.146T + 83T^{2} \) |
| 89 | \( 1 + 1.54T + 89T^{2} \) |
| 97 | \( 1 + 7.32T + 97T^{2} \) |
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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
−10.19270714515967395864302102494, −8.738697774736115932429951629269, −8.218317865691176635698276218673, −7.18591162469180861962059739841, −6.45231649340486452757770118428, −5.34619381793888950509502531766, −4.39405183658322571128944608155, −3.93389707846361486050084934348, −2.68337922048314293816474941490, −2.06747234374865495005375723467,
2.06747234374865495005375723467, 2.68337922048314293816474941490, 3.93389707846361486050084934348, 4.39405183658322571128944608155, 5.34619381793888950509502531766, 6.45231649340486452757770118428, 7.18591162469180861962059739841, 8.218317865691176635698276218673, 8.738697774736115932429951629269, 10.19270714515967395864302102494
