| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 15.9·5-s − 6·6-s − 18.6·7-s + 8·8-s + 9·9-s + 31.9·10-s − 2.67·11-s − 12·12-s + 79.1·13-s − 37.3·14-s − 47.9·15-s + 16·16-s + 50.9·17-s + 18·18-s − 59.4·19-s + 63.9·20-s + 56.0·21-s − 5.34·22-s − 34.6·23-s − 24·24-s + 130.·25-s + 158.·26-s − 27·27-s − 74.7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.42·5-s − 0.408·6-s − 1.00·7-s + 0.353·8-s + 0.333·9-s + 1.01·10-s − 0.0732·11-s − 0.288·12-s + 1.68·13-s − 0.713·14-s − 0.825·15-s + 0.250·16-s + 0.726·17-s + 0.235·18-s − 0.718·19-s + 0.714·20-s + 0.582·21-s − 0.0517·22-s − 0.313·23-s − 0.204·24-s + 1.04·25-s + 1.19·26-s − 0.192·27-s − 0.504·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.073213143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.073213143\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 59 | \( 1 - 59T \) |
| good | 5 | \( 1 - 15.9T + 125T^{2} \) |
| 7 | \( 1 + 18.6T + 343T^{2} \) |
| 11 | \( 1 + 2.67T + 1.33e3T^{2} \) |
| 13 | \( 1 - 79.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 59.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 34.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 35.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 281.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 77.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 27.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 80.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 178.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 58.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 287.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 765.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 568.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 594.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 100.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 122.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 978.T + 9.12e5T^{2} \) |
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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
−11.01350425943980129545773263488, −10.18754622412268757768198090475, −9.520926239342830872762371262371, −8.215718443296697468568429548076, −6.46034156152692299415959155117, −6.28770065835165885960493014785, −5.38268902547389256963234056544, −3.98459940382262151135628848689, −2.68553988297879590658569151272, −1.20258724379566111144012170101,
1.20258724379566111144012170101, 2.68553988297879590658569151272, 3.98459940382262151135628848689, 5.38268902547389256963234056544, 6.28770065835165885960493014785, 6.46034156152692299415959155117, 8.215718443296697468568429548076, 9.520926239342830872762371262371, 10.18754622412268757768198090475, 11.01350425943980129545773263488
