| L(s) = 1 | + 3.65·2-s − 3·3-s + 5.38·4-s − 2.03·5-s − 10.9·6-s − 28.9·7-s − 9.55·8-s + 9·9-s − 7.43·10-s − 11·11-s − 16.1·12-s − 48.1·13-s − 105.·14-s + 6.09·15-s − 78.0·16-s − 35.3·17-s + 32.9·18-s + 101.·19-s − 10.9·20-s + 86.9·21-s − 40.2·22-s − 27.1·23-s + 28.6·24-s − 120.·25-s − 176.·26-s − 27·27-s − 156.·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.577·3-s + 0.673·4-s − 0.181·5-s − 0.746·6-s − 1.56·7-s − 0.422·8-s + 0.333·9-s − 0.234·10-s − 0.301·11-s − 0.388·12-s − 1.02·13-s − 2.02·14-s + 0.104·15-s − 1.21·16-s − 0.504·17-s + 0.431·18-s + 1.22·19-s − 0.122·20-s + 0.903·21-s − 0.390·22-s − 0.245·23-s + 0.243·24-s − 0.967·25-s − 1.32·26-s − 0.192·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9532130223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9532130223\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 3.65T + 8T^{2} \) |
| 5 | \( 1 + 2.03T + 125T^{2} \) |
| 7 | \( 1 + 28.9T + 343T^{2} \) |
| 13 | \( 1 + 48.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 107.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 23.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 286.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 428.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 122.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 415.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 418.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 594.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 226.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.32T + 3.89e5T^{2} \) |
| 79 | \( 1 - 593.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 94.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 202.T + 9.12e5T^{2} \) |
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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
−8.995727695924273121028773958121, −7.67558042831216342182030467206, −6.85413333461019533851678028785, −6.31037356774987380486146533685, −5.43549883612581668073671477858, −4.90176812565028008386142676447, −3.81064923210590699129174097890, −3.24144706235750881148176416807, −2.24600503108116751297117196077, −0.35231681230005567736394417189,
0.35231681230005567736394417189, 2.24600503108116751297117196077, 3.24144706235750881148176416807, 3.81064923210590699129174097890, 4.90176812565028008386142676447, 5.43549883612581668073671477858, 6.31037356774987380486146533685, 6.85413333461019533851678028785, 7.67558042831216342182030467206, 8.995727695924273121028773958121
