| L(s) = 1 | − 1.61·2-s − 0.965·3-s + 0.594·4-s + 4.40·5-s + 1.55·6-s − 3.07·7-s + 2.26·8-s − 2.06·9-s − 7.10·10-s + 0.281·11-s − 0.574·12-s + 13-s + 4.95·14-s − 4.25·15-s − 4.83·16-s − 6.69·17-s + 3.33·18-s + 6.10·19-s + 2.62·20-s + 2.97·21-s − 0.453·22-s + 4.48·23-s − 2.18·24-s + 14.4·25-s − 1.61·26-s + 4.89·27-s − 1.82·28-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 0.557·3-s + 0.297·4-s + 1.97·5-s + 0.635·6-s − 1.16·7-s + 0.800·8-s − 0.689·9-s − 2.24·10-s + 0.0849·11-s − 0.165·12-s + 0.277·13-s + 1.32·14-s − 1.09·15-s − 1.20·16-s − 1.62·17-s + 0.784·18-s + 1.39·19-s + 0.585·20-s + 0.648·21-s − 0.0967·22-s + 0.934·23-s − 0.446·24-s + 2.88·25-s − 0.315·26-s + 0.941·27-s − 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7977453081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7977453081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 0.965T + 3T^{2} \) |
| 5 | \( 1 - 4.40T + 5T^{2} \) |
| 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 - 0.281T + 11T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 - 6.10T + 19T^{2} \) |
| 23 | \( 1 - 4.48T + 23T^{2} \) |
| 29 | \( 1 - 5.72T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 - 5.59T + 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 + 7.99T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 - 2.73T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 4.20T + 79T^{2} \) |
| 83 | \( 1 + 8.91T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 4.64T + 97T^{2} \) |
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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
−9.484972386184390078967988036088, −9.138505631688928258597889153842, −8.394503301862515387177489329389, −6.74117686577329736747347233397, −6.67064237183892004161617933236, −5.56644064567415350580374848387, −4.90700808359586869707891812148, −3.12199474007869930497130361361, −2.07957131727997939328981993813, −0.796935545574333260606733648440,
0.796935545574333260606733648440, 2.07957131727997939328981993813, 3.12199474007869930497130361361, 4.90700808359586869707891812148, 5.56644064567415350580374848387, 6.67064237183892004161617933236, 6.74117686577329736747347233397, 8.394503301862515387177489329389, 9.138505631688928258597889153842, 9.484972386184390078967988036088
