| L(s) = 1 | − 1.93·2-s + 3-s + 1.76·4-s + 0.142·5-s − 1.93·6-s + 3.99·7-s + 0.462·8-s + 9-s − 0.276·10-s + 2.93·11-s + 1.76·12-s − 13-s − 7.74·14-s + 0.142·15-s − 4.42·16-s − 5.84·17-s − 1.93·18-s + 6.34·19-s + 0.250·20-s + 3.99·21-s − 5.69·22-s − 4.92·23-s + 0.462·24-s − 4.97·25-s + 1.93·26-s + 27-s + 7.03·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.577·3-s + 0.880·4-s + 0.0637·5-s − 0.791·6-s + 1.51·7-s + 0.163·8-s + 0.333·9-s − 0.0873·10-s + 0.885·11-s + 0.508·12-s − 0.277·13-s − 2.07·14-s + 0.0367·15-s − 1.10·16-s − 1.41·17-s − 0.457·18-s + 1.45·19-s + 0.0561·20-s + 0.871·21-s − 1.21·22-s − 1.02·23-s + 0.0944·24-s − 0.995·25-s + 0.380·26-s + 0.192·27-s + 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.485543889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.485543889\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 5 | \( 1 - 0.142T + 5T^{2} \) |
| 7 | \( 1 - 3.99T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 17 | \( 1 + 5.84T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 0.367T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 - 3.03T + 41T^{2} \) |
| 43 | \( 1 - 3.61T + 43T^{2} \) |
| 47 | \( 1 - 6.96T + 47T^{2} \) |
| 53 | \( 1 - 4.59T + 53T^{2} \) |
| 59 | \( 1 + 6.80T + 59T^{2} \) |
| 61 | \( 1 - 8.96T + 61T^{2} \) |
| 67 | \( 1 - 9.34T + 67T^{2} \) |
| 71 | \( 1 + 8.00T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 6.54T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.365792273669213648919137278562, −7.967965407250899920694282961784, −7.36437868793243276802616029950, −6.61862500295807505281027978604, −5.49453472091219590067540025908, −4.50281891442161521853298689738, −3.96453609102862513977809199662, −2.41340569460972559592407962311, −1.80617241757714875595077096579, −0.885820646933616303034278975476,
0.885820646933616303034278975476, 1.80617241757714875595077096579, 2.41340569460972559592407962311, 3.96453609102862513977809199662, 4.50281891442161521853298689738, 5.49453472091219590067540025908, 6.61862500295807505281027978604, 7.36437868793243276802616029950, 7.967965407250899920694282961784, 8.365792273669213648919137278562
