| L(s) = 1 | − 1.51·2-s + 0.630·3-s + 0.305·4-s − 0.956·6-s + 7-s + 2.57·8-s − 2.60·9-s + 4.75·11-s + 0.192·12-s + 2.10·13-s − 1.51·14-s − 4.51·16-s − 5.16·17-s + 3.95·18-s + 5.33·19-s + 0.630·21-s − 7.21·22-s + 23-s + 1.62·24-s − 3.19·26-s − 3.53·27-s + 0.305·28-s − 5.93·29-s + 1.56·31-s + 1.71·32-s + 2.99·33-s + 7.83·34-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.363·3-s + 0.152·4-s − 0.390·6-s + 0.377·7-s + 0.909·8-s − 0.867·9-s + 1.43·11-s + 0.0556·12-s + 0.583·13-s − 0.405·14-s − 1.12·16-s − 1.25·17-s + 0.931·18-s + 1.22·19-s + 0.137·21-s − 1.53·22-s + 0.208·23-s + 0.330·24-s − 0.626·26-s − 0.679·27-s + 0.0578·28-s − 1.10·29-s + 0.280·31-s + 0.303·32-s + 0.521·33-s + 1.34·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.209579940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209579940\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.51T + 2T^{2} \) |
| 3 | \( 1 - 0.630T + 3T^{2} \) |
| 11 | \( 1 - 4.75T + 11T^{2} \) |
| 13 | \( 1 - 2.10T + 13T^{2} \) |
| 17 | \( 1 + 5.16T + 17T^{2} \) |
| 19 | \( 1 - 5.33T + 19T^{2} \) |
| 29 | \( 1 + 5.93T + 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 - 8.19T + 37T^{2} \) |
| 41 | \( 1 - 6.11T + 41T^{2} \) |
| 43 | \( 1 - 4.23T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 4.68T + 53T^{2} \) |
| 59 | \( 1 + 6.62T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 5.37T + 67T^{2} \) |
| 71 | \( 1 + 1.79T + 71T^{2} \) |
| 73 | \( 1 - 4.69T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 + 9.67T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.724743837093171414162046310115, −7.78312590402871456388118611522, −7.36006703923416772886282820684, −6.34118088975193246080794632997, −5.64006443955380362428139624133, −4.47469587297478693340134010889, −3.90800329772359242898936012663, −2.74965551815177381599642881159, −1.69144712667100677822749591640, −0.77008632289525629250303105000,
0.77008632289525629250303105000, 1.69144712667100677822749591640, 2.74965551815177381599642881159, 3.90800329772359242898936012663, 4.47469587297478693340134010889, 5.64006443955380362428139624133, 6.34118088975193246080794632997, 7.36006703923416772886282820684, 7.78312590402871456388118611522, 8.724743837093171414162046310115
