| L(s) = 1 | + 3.33·2-s − 3·3-s + 3.14·4-s − 10.0·6-s + 7·7-s − 16.2·8-s + 9·9-s + 18.3·11-s − 9.42·12-s + 10.1·13-s + 23.3·14-s − 79.2·16-s − 24.6·17-s + 30.0·18-s + 77.4·19-s − 21·21-s + 61.1·22-s + 149.·23-s + 48.6·24-s + 33.8·26-s − 27·27-s + 21.9·28-s − 10.2·29-s + 124.·31-s − 134.·32-s − 54.9·33-s − 82.4·34-s + ⋯ |
| L(s) = 1 | + 1.18·2-s − 0.577·3-s + 0.392·4-s − 0.681·6-s + 0.377·7-s − 0.716·8-s + 0.333·9-s + 0.502·11-s − 0.226·12-s + 0.216·13-s + 0.446·14-s − 1.23·16-s − 0.352·17-s + 0.393·18-s + 0.934·19-s − 0.218·21-s + 0.592·22-s + 1.35·23-s + 0.413·24-s + 0.255·26-s − 0.192·27-s + 0.148·28-s − 0.0659·29-s + 0.720·31-s − 0.744·32-s − 0.290·33-s − 0.415·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.957863831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957863831\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 3.33T + 8T^{2} \) |
| 11 | \( 1 - 18.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 495.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 220.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 532.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 324.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 653.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 819.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 466.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 173.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 810.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 12.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 33.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 810.T + 9.12e5T^{2} \) |
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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
−10.82030043092246493745505311062, −9.562347740424404653863932800412, −8.776331212603078900572116288308, −7.44307353220930852039362628418, −6.46972341641155003054026045959, −5.61653867899764130991224927091, −4.78188530861677088519054503989, −3.94696473929894485030788072505, −2.72019472199818740806466589942, −0.953974473112225349203123265015,
0.953974473112225349203123265015, 2.72019472199818740806466589942, 3.94696473929894485030788072505, 4.78188530861677088519054503989, 5.61653867899764130991224927091, 6.46972341641155003054026045959, 7.44307353220930852039362628418, 8.776331212603078900572116288308, 9.562347740424404653863932800412, 10.82030043092246493745505311062
