| L(s) = 1 | + 25·5-s − 223.·7-s − 685.·11-s + 125.·13-s − 475.·17-s − 2.44e3·19-s − 346.·23-s + 625·25-s − 4.34e3·29-s + 2.78e3·31-s − 5.59e3·35-s − 5.06e3·37-s + 3.76e3·41-s + 1.63e4·43-s − 1.58e4·47-s + 3.32e4·49-s − 1.96e4·53-s − 1.71e4·55-s + 3.54e4·59-s + 1.89e4·61-s + 3.14e3·65-s − 4.87e4·67-s + 7.69e4·71-s + 4.33e4·73-s + 1.53e5·77-s − 7.96e4·79-s + 674.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.72·7-s − 1.70·11-s + 0.206·13-s − 0.398·17-s − 1.55·19-s − 0.136·23-s + 0.200·25-s − 0.960·29-s + 0.520·31-s − 0.772·35-s − 0.608·37-s + 0.349·41-s + 1.34·43-s − 1.04·47-s + 1.98·49-s − 0.958·53-s − 0.763·55-s + 1.32·59-s + 0.651·61-s + 0.0924·65-s − 1.32·67-s + 1.81·71-s + 0.952·73-s + 2.94·77-s − 1.43·79-s + 0.0107·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6632703016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6632703016\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 7 | \( 1 + 223.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 685.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 125.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 475.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.44e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 346.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.34e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.06e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.76e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.63e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.54e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 674.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.21e4T + 8.58e9T^{2} \) |
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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
−9.746978259313233038757508733217, −8.897013003502564056854845800386, −7.950219466818259094684483133979, −6.84998734049395645569352196508, −6.16298839822493153152094691114, −5.32220585279314545641806314533, −4.05080243673744788987092125262, −2.93625416663438145677146564340, −2.17437641179912125021763040671, −0.35123185088463889541935502813,
0.35123185088463889541935502813, 2.17437641179912125021763040671, 2.93625416663438145677146564340, 4.05080243673744788987092125262, 5.32220585279314545641806314533, 6.16298839822493153152094691114, 6.84998734049395645569352196508, 7.950219466818259094684483133979, 8.897013003502564056854845800386, 9.746978259313233038757508733217
