| L(s) = 1 | − 2.04·2-s − 3.82·4-s − 12.0·5-s + 29.7·7-s + 24.1·8-s + 24.6·10-s − 28.0·11-s − 60.7·14-s − 18.7·16-s + 50.6·17-s + 105.·19-s + 46.2·20-s + 57.3·22-s + 160.·23-s + 20.9·25-s − 113.·28-s − 140.·29-s + 223.·31-s − 155.·32-s − 103.·34-s − 359.·35-s − 228.·37-s − 214.·38-s − 291.·40-s + 295.·41-s + 192.·43-s + 107.·44-s + ⋯ |
| L(s) = 1 | − 0.722·2-s − 0.478·4-s − 1.08·5-s + 1.60·7-s + 1.06·8-s + 0.780·10-s − 0.769·11-s − 1.15·14-s − 0.292·16-s + 0.722·17-s + 1.26·19-s + 0.517·20-s + 0.555·22-s + 1.45·23-s + 0.167·25-s − 0.768·28-s − 0.897·29-s + 1.29·31-s − 0.856·32-s − 0.521·34-s − 1.73·35-s − 1.01·37-s − 0.916·38-s − 1.15·40-s + 1.12·41-s + 0.681·43-s + 0.368·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.192631970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192631970\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.04T + 8T^{2} \) |
| 5 | \( 1 + 12.0T + 125T^{2} \) |
| 7 | \( 1 - 29.7T + 343T^{2} \) |
| 11 | \( 1 + 28.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 50.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 228.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 36.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 438.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 286.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 537.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 75.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 17.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 334.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 748.T + 9.12e5T^{2} \) |
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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
−8.909085508199835528114261839762, −8.242499145991333519200202498587, −7.58172207529768478435641050755, −7.38012784123140130079362713399, −5.51693701392394956026933216455, −4.90072886677327140342850509481, −4.18000041163567125977808027701, −3.03076895333184659764406974593, −1.50020161056778332147490104735, −0.65912035406129351524050767619,
0.65912035406129351524050767619, 1.50020161056778332147490104735, 3.03076895333184659764406974593, 4.18000041163567125977808027701, 4.90072886677327140342850509481, 5.51693701392394956026933216455, 7.38012784123140130079362713399, 7.58172207529768478435641050755, 8.242499145991333519200202498587, 8.909085508199835528114261839762
