| L(s) = 1 | − 5.78·2-s + 1.45·4-s − 25·5-s + 17.6·7-s + 176.·8-s + 144.·10-s − 121·11-s + 674.·13-s − 102.·14-s − 1.06e3·16-s + 2.11e3·17-s − 2.30e3·19-s − 36.4·20-s + 699.·22-s + 3.07e3·23-s + 625·25-s − 3.90e3·26-s + 25.7·28-s + 1.43e3·29-s − 5.15e3·31-s + 526.·32-s − 1.22e4·34-s − 441.·35-s + 6.92e3·37-s + 1.33e4·38-s − 4.41e3·40-s − 2.84e3·41-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.0455·4-s − 0.447·5-s + 0.136·7-s + 0.975·8-s + 0.457·10-s − 0.301·11-s + 1.10·13-s − 0.139·14-s − 1.04·16-s + 1.77·17-s − 1.46·19-s − 0.0203·20-s + 0.308·22-s + 1.21·23-s + 0.200·25-s − 1.13·26-s + 0.00619·28-s + 0.317·29-s − 0.963·31-s + 0.0909·32-s − 1.81·34-s − 0.0608·35-s + 0.832·37-s + 1.49·38-s − 0.436·40-s − 0.264·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9810648689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9810648689\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 5.78T + 32T^{2} \) |
| 7 | \( 1 - 17.6T + 1.68e4T^{2} \) |
| 13 | \( 1 - 674.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.11e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.30e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.45e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.97e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 378.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.20e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.12e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.52e5T + 8.58e9T^{2} \) |
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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
−10.16635608756747259940231223861, −9.119565993894597502881696095824, −8.383053286391489149839130517579, −7.77979711033930729970942349196, −6.75875940176293052714813464676, −5.48942556043060736269082089962, −4.36400482999512249024689398846, −3.26354768190471446030513561848, −1.60921500684387199888985101616, −0.62377404497088851152038135075,
0.62377404497088851152038135075, 1.60921500684387199888985101616, 3.26354768190471446030513561848, 4.36400482999512249024689398846, 5.48942556043060736269082089962, 6.75875940176293052714813464676, 7.77979711033930729970942349196, 8.383053286391489149839130517579, 9.119565993894597502881696095824, 10.16635608756747259940231223861
