| L(s) = 1 | − 2.63·3-s − 2.05·5-s + 4.98·7-s + 3.93·9-s + 0.504·11-s + 4.69·13-s + 5.40·15-s + 5.45·17-s + 5.31·19-s − 13.1·21-s + 4.77·23-s − 0.792·25-s − 2.46·27-s − 5.28·29-s + 9.88·31-s − 1.32·33-s − 10.2·35-s + 7.71·37-s − 12.3·39-s − 4.02·41-s + 7.33·43-s − 8.07·45-s + 8.07·47-s + 17.8·49-s − 14.3·51-s − 7.42·53-s − 1.03·55-s + ⋯ |
| L(s) = 1 | − 1.52·3-s − 0.917·5-s + 1.88·7-s + 1.31·9-s + 0.152·11-s + 1.30·13-s + 1.39·15-s + 1.32·17-s + 1.21·19-s − 2.86·21-s + 0.995·23-s − 0.158·25-s − 0.474·27-s − 0.980·29-s + 1.77·31-s − 0.231·33-s − 1.72·35-s + 1.26·37-s − 1.97·39-s − 0.628·41-s + 1.11·43-s − 1.20·45-s + 1.17·47-s + 2.55·49-s − 2.01·51-s − 1.02·53-s − 0.139·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526008666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526008666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 2.63T + 3T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 7 | \( 1 - 4.98T + 7T^{2} \) |
| 11 | \( 1 - 0.504T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 - 5.31T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 - 9.88T + 31T^{2} \) |
| 37 | \( 1 - 7.71T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 - 7.33T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 + 7.42T + 53T^{2} \) |
| 59 | \( 1 + 8.76T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 - 7.49T + 67T^{2} \) |
| 71 | \( 1 - 0.654T + 71T^{2} \) |
| 73 | \( 1 + 8.43T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 7.92T + 97T^{2} \) |
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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
−8.159311839156434395683817481368, −7.72433058624705934936274384962, −7.09208507226955891617349827007, −5.88766726732163066988296614597, −5.60116452078879408572635460212, −4.67434161935533550000753079930, −4.23030430054808906747537237381, −3.12053322888079208289733136533, −1.35130210449271309840014062720, −0.939825221351275864457174755716,
0.939825221351275864457174755716, 1.35130210449271309840014062720, 3.12053322888079208289733136533, 4.23030430054808906747537237381, 4.67434161935533550000753079930, 5.60116452078879408572635460212, 5.88766726732163066988296614597, 7.09208507226955891617349827007, 7.72433058624705934936274384962, 8.159311839156434395683817481368
