| L(s) = 1 | + 17.5·3-s + 32.5·5-s + 220.·7-s + 64.1·9-s + 85.7·11-s + 1.03e3·13-s + 570.·15-s − 313.·17-s − 458.·19-s + 3.86e3·21-s + 3.44e3·23-s − 2.06e3·25-s − 3.13e3·27-s − 841·29-s + 7.98e3·31-s + 1.50e3·33-s + 7.19e3·35-s + 152.·37-s + 1.81e4·39-s − 1.84e4·41-s − 2.07e3·43-s + 2.08e3·45-s − 1.58e4·47-s + 3.19e4·49-s − 5.48e3·51-s + 9.24e3·53-s + 2.79e3·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s + 0.582·5-s + 1.70·7-s + 0.264·9-s + 0.213·11-s + 1.69·13-s + 0.654·15-s − 0.262·17-s − 0.291·19-s + 1.91·21-s + 1.35·23-s − 0.660·25-s − 0.827·27-s − 0.185·29-s + 1.49·31-s + 0.240·33-s + 0.992·35-s + 0.0183·37-s + 1.90·39-s − 1.71·41-s − 0.170·43-s + 0.153·45-s − 1.04·47-s + 1.90·49-s − 0.295·51-s + 0.451·53-s + 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.988665861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.988665861\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 841T \) |
| good | 3 | \( 1 - 17.5T + 243T^{2} \) |
| 5 | \( 1 - 32.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 220.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 85.7T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 313.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 458.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.44e3T + 6.43e6T^{2} \) |
| 31 | \( 1 - 7.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 152.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.07e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.43e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.95e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.19e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.93e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.36e4T + 8.58e9T^{2} \) |
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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
−10.21145234171246143687941548355, −8.988217075716532285560683587107, −8.509174173644835508663995804617, −7.87912066614953416650748676814, −6.58290572169929495855465011326, −5.44952451680479356322422811518, −4.37072360387744897767862066721, −3.23636283447365096600782409185, −1.99793873762039358751603309262, −1.25121834703043455695353606724,
1.25121834703043455695353606724, 1.99793873762039358751603309262, 3.23636283447365096600782409185, 4.37072360387744897767862066721, 5.44952451680479356322422811518, 6.58290572169929495855465011326, 7.87912066614953416650748676814, 8.509174173644835508663995804617, 8.988217075716532285560683587107, 10.21145234171246143687941548355
