| L(s) = 1 | + 3·3-s − 4.87·5-s − 25.6·7-s + 9·9-s + 59.7·11-s + 13·13-s − 14.6·15-s − 75.4·17-s − 116.·19-s − 76.8·21-s − 90.5·23-s − 101.·25-s + 27·27-s − 187.·29-s − 225.·31-s + 179.·33-s + 124.·35-s + 290.·37-s + 39·39-s − 191.·41-s + 326.·43-s − 43.8·45-s − 406.·47-s + 313.·49-s − 226.·51-s − 426.·53-s − 291.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.436·5-s − 1.38·7-s + 0.333·9-s + 1.63·11-s + 0.277·13-s − 0.251·15-s − 1.07·17-s − 1.40·19-s − 0.798·21-s − 0.820·23-s − 0.809·25-s + 0.192·27-s − 1.19·29-s − 1.30·31-s + 0.945·33-s + 0.603·35-s + 1.29·37-s + 0.160·39-s − 0.728·41-s + 1.15·43-s − 0.145·45-s − 1.26·47-s + 0.912·49-s − 0.621·51-s − 1.10·53-s − 0.714·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.87T + 125T^{2} \) |
| 7 | \( 1 + 25.6T + 343T^{2} \) |
| 11 | \( 1 - 59.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 75.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 90.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 191.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 406.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 426.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 524.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 968.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 8.39T + 3.57e5T^{2} \) |
| 73 | \( 1 + 903.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 952.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 90.7T + 9.12e5T^{2} \) |
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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
−10.79479718180286426519951884760, −9.472805026810281246778095833868, −9.147686717995460193798278020972, −7.965347053620023151782253387617, −6.72377644769637053377010965107, −6.17343318526163125708986232958, −4.14240725570613377864073640081, −3.60831962071639935710322882913, −2.00893613230940240624886554959, 0,
2.00893613230940240624886554959, 3.60831962071639935710322882913, 4.14240725570613377864073640081, 6.17343318526163125708986232958, 6.72377644769637053377010965107, 7.965347053620023151782253387617, 9.147686717995460193798278020972, 9.472805026810281246778095833868, 10.79479718180286426519951884760
