| L(s) = 1 | + 2.08·2-s + 2.34·4-s + 0.992·5-s + 2.45·7-s + 0.726·8-s + 2.06·10-s + 0.745·11-s + 0.730·13-s + 5.11·14-s − 3.18·16-s − 4.76·17-s + 1.38·19-s + 2.33·20-s + 1.55·22-s + 5.31·23-s − 4.01·25-s + 1.52·26-s + 5.75·28-s − 0.563·31-s − 8.08·32-s − 9.93·34-s + 2.43·35-s + 9.49·37-s + 2.88·38-s + 0.721·40-s + 5.63·41-s + 11.0·43-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.17·4-s + 0.443·5-s + 0.926·7-s + 0.256·8-s + 0.654·10-s + 0.224·11-s + 0.202·13-s + 1.36·14-s − 0.795·16-s − 1.15·17-s + 0.317·19-s + 0.521·20-s + 0.331·22-s + 1.10·23-s − 0.803·25-s + 0.298·26-s + 1.08·28-s − 0.101·31-s − 1.42·32-s − 1.70·34-s + 0.411·35-s + 1.56·37-s + 0.467·38-s + 0.114·40-s + 0.880·41-s + 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.890301869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.890301869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 0.992T + 5T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 - 0.745T + 11T^{2} \) |
| 13 | \( 1 - 0.730T + 13T^{2} \) |
| 17 | \( 1 + 4.76T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 - 5.31T + 23T^{2} \) |
| 31 | \( 1 + 0.563T + 31T^{2} \) |
| 37 | \( 1 - 9.49T + 37T^{2} \) |
| 41 | \( 1 - 5.63T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 - 8.54T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 6.10T + 71T^{2} \) |
| 73 | \( 1 - 5.41T + 73T^{2} \) |
| 79 | \( 1 + 7.67T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 9.90T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.60134780594485029406262109264, −7.03248239979068281216089447343, −6.18849755153329320041791895990, −5.67578376742198882956753213847, −5.03918744605820972537825817952, −4.27275011037040106862817940095, −3.89936414732418839019875261911, −2.65278016016051110062197805639, −2.23701910420667289000917495422, −1.00693681199066280822759296783,
1.00693681199066280822759296783, 2.23701910420667289000917495422, 2.65278016016051110062197805639, 3.89936414732418839019875261911, 4.27275011037040106862817940095, 5.03918744605820972537825817952, 5.67578376742198882956753213847, 6.18849755153329320041791895990, 7.03248239979068281216089447343, 7.60134780594485029406262109264
