| L(s) = 1 | − 3·3-s − 7.29·5-s − 5.87·7-s + 9·9-s + 51.1·11-s − 13·13-s + 21.8·15-s − 73.7·17-s − 59.9·19-s + 17.6·21-s − 69.8·23-s − 71.8·25-s − 27·27-s − 294.·29-s + 334.·31-s − 153.·33-s + 42.8·35-s + 261.·37-s + 39·39-s + 222.·41-s − 79.2·43-s − 65.6·45-s + 584.·47-s − 308.·49-s + 221.·51-s + 465.·53-s − 373.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.652·5-s − 0.317·7-s + 0.333·9-s + 1.40·11-s − 0.277·13-s + 0.376·15-s − 1.05·17-s − 0.723·19-s + 0.183·21-s − 0.633·23-s − 0.574·25-s − 0.192·27-s − 1.88·29-s + 1.93·31-s − 0.809·33-s + 0.206·35-s + 1.16·37-s + 0.160·39-s + 0.848·41-s − 0.281·43-s − 0.217·45-s + 1.81·47-s − 0.899·49-s + 0.607·51-s + 1.20·53-s − 0.914·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.118425731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118425731\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 7.29T + 125T^{2} \) |
| 7 | \( 1 + 5.87T + 343T^{2} \) |
| 11 | \( 1 - 51.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 73.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 59.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 69.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 294.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 334.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 261.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 79.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 584.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 465.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 548.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 384.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 307.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 844.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 30.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + 19.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 513.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 787.T + 9.12e5T^{2} \) |
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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
−10.20891073958131807103078334760, −9.387565959722161919265826529864, −8.500006357915870705092934633128, −7.42211414701795051215366631170, −6.57084331338770876493029489437, −5.83817513496424967137746864135, −4.36269930782292008735014194048, −3.89474623111073750624978943301, −2.20250858177020550000752140290, −0.63368191710452989362268504219,
0.63368191710452989362268504219, 2.20250858177020550000752140290, 3.89474623111073750624978943301, 4.36269930782292008735014194048, 5.83817513496424967137746864135, 6.57084331338770876493029489437, 7.42211414701795051215366631170, 8.500006357915870705092934633128, 9.387565959722161919265826529864, 10.20891073958131807103078334760
