| L(s) = 1 | − 2.36·3-s − 1.32·5-s − 4.19·7-s + 2.60·9-s + 3.11·11-s − 0.743·13-s + 3.12·15-s − 3.53·17-s − 8.40·19-s + 9.92·21-s + 5.22·23-s − 3.25·25-s + 0.937·27-s − 2.06·29-s − 8.35·31-s − 7.38·33-s + 5.53·35-s − 8.88·37-s + 1.75·39-s − 3.87·41-s − 2.84·43-s − 3.43·45-s − 4.01·47-s + 10.5·49-s + 8.36·51-s + 6.87·53-s − 4.11·55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s − 0.590·5-s − 1.58·7-s + 0.868·9-s + 0.940·11-s − 0.206·13-s + 0.806·15-s − 0.856·17-s − 1.92·19-s + 2.16·21-s + 1.08·23-s − 0.651·25-s + 0.180·27-s − 0.383·29-s − 1.50·31-s − 1.28·33-s + 0.935·35-s − 1.46·37-s + 0.281·39-s − 0.604·41-s − 0.433·43-s − 0.512·45-s − 0.585·47-s + 1.51·49-s + 1.17·51-s + 0.943·53-s − 0.555·55-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
| \(L(1)\) |
\(\approx\) |
\(0.09197686999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09197686999\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.36T + 3T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 7 | \( 1 + 4.19T + 7T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 + 0.743T + 13T^{2} \) |
| 17 | \( 1 + 3.53T + 17T^{2} \) |
| 19 | \( 1 + 8.40T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 + 8.35T + 31T^{2} \) |
| 37 | \( 1 + 8.88T + 37T^{2} \) |
| 41 | \( 1 + 3.87T + 41T^{2} \) |
| 43 | \( 1 + 2.84T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 - 6.87T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 6.26T + 61T^{2} \) |
| 67 | \( 1 + 5.24T + 67T^{2} \) |
| 71 | \( 1 + 7.25T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 6.87T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.751971035331726250786292077129, −7.32257726624000966020100772934, −6.77816501770776827919024092495, −6.33009509694633836781114173971, −5.66481684669054463558917712427, −4.65253152131716704085743703614, −3.95514378285794200819423240062, −3.17044000920957862510058006825, −1.79091779406791301066708261385, −0.17838750109957839735146703485,
0.17838750109957839735146703485, 1.79091779406791301066708261385, 3.17044000920957862510058006825, 3.95514378285794200819423240062, 4.65253152131716704085743703614, 5.66481684669054463558917712427, 6.33009509694633836781114173971, 6.77816501770776827919024092495, 7.32257726624000966020100772934, 8.751971035331726250786292077129
