| L(s) = 1 | + 5.16·3-s + 9·5-s − 109.·7-s − 216.·9-s + 121·11-s − 390.·13-s + 46.4·15-s + 422.·17-s − 2.24e3·19-s − 567.·21-s − 217.·23-s − 3.04e3·25-s − 2.37e3·27-s − 4.43e3·29-s + 4.80e3·31-s + 625.·33-s − 988.·35-s − 1.27e4·37-s − 2.01e3·39-s − 7.01e3·41-s + 7.70e3·43-s − 1.94e3·45-s + 1.57e4·47-s − 4.74e3·49-s + 2.18e3·51-s + 1.22e4·53-s + 1.08e3·55-s + ⋯ |
| L(s) = 1 | + 0.331·3-s + 0.160·5-s − 0.847·7-s − 0.890·9-s + 0.301·11-s − 0.640·13-s + 0.0533·15-s + 0.354·17-s − 1.42·19-s − 0.280·21-s − 0.0856·23-s − 0.974·25-s − 0.626·27-s − 0.979·29-s + 0.898·31-s + 0.0999·33-s − 0.136·35-s − 1.53·37-s − 0.212·39-s − 0.651·41-s + 0.635·43-s − 0.143·45-s + 1.04·47-s − 0.282·49-s + 0.117·51-s + 0.599·53-s + 0.0485·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 - 5.16T + 243T^{2} \) |
| 5 | \( 1 - 9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 109.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 390.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 422.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 217.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.27e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.70e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.57e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.37e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.97e4T + 8.58e9T^{2} \) |
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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
−12.71104090296938544698474588718, −11.68072538132719985891178053807, −10.32919446343358559879481518266, −9.308447901155435725996407038845, −8.243418962414621589370502451663, −6.78275221532505462244127429013, −5.59424691102420022147614481185, −3.77571289706663469526055859700, −2.34191600041547411065664860379, 0,
2.34191600041547411065664860379, 3.77571289706663469526055859700, 5.59424691102420022147614481185, 6.78275221532505462244127429013, 8.243418962414621589370502451663, 9.308447901155435725996407038845, 10.32919446343358559879481518266, 11.68072538132719985891178053807, 12.71104090296938544698474588718
