| L(s) = 1 | − 0.864·3-s + 2.57·5-s − 20.6·7-s − 26.2·9-s − 13.0·11-s + 52.3·13-s − 2.22·15-s − 95.5·17-s − 132.·19-s + 17.8·21-s − 76.3·23-s − 118.·25-s + 46.0·27-s + 193.·29-s − 31·31-s + 11.3·33-s − 53.0·35-s − 175.·37-s − 45.2·39-s + 517.·41-s + 435.·43-s − 67.5·45-s − 314.·47-s + 81.8·49-s + 82.6·51-s + 129.·53-s − 33.6·55-s + ⋯ |
| L(s) = 1 | − 0.166·3-s + 0.230·5-s − 1.11·7-s − 0.972·9-s − 0.358·11-s + 1.11·13-s − 0.0383·15-s − 1.36·17-s − 1.59·19-s + 0.185·21-s − 0.692·23-s − 0.947·25-s + 0.328·27-s + 1.24·29-s − 0.179·31-s + 0.0597·33-s − 0.256·35-s − 0.781·37-s − 0.185·39-s + 1.97·41-s + 1.54·43-s − 0.223·45-s − 0.975·47-s + 0.238·49-s + 0.226·51-s + 0.336·53-s − 0.0826·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + 31T \) |
| good | 3 | \( 1 + 0.864T + 27T^{2} \) |
| 5 | \( 1 - 2.57T + 125T^{2} \) |
| 7 | \( 1 + 20.6T + 343T^{2} \) |
| 11 | \( 1 + 13.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 132.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 193.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 517.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 435.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 129.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 378.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 586.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 326.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 924.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 400.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 218.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 741.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.45e3T + 9.12e5T^{2} \) |
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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
−12.57861362048066741077805147256, −11.25963686993249342239542037875, −10.44919490321444497441013971689, −9.156533760745363151738941361763, −8.291493259112238885101107706350, −6.52793379395203378227019374381, −5.91136463572116904130048066593, −4.08780551203771438749953993296, −2.52198549012378428953338608114, 0,
2.52198549012378428953338608114, 4.08780551203771438749953993296, 5.91136463572116904130048066593, 6.52793379395203378227019374381, 8.291493259112238885101107706350, 9.156533760745363151738941361763, 10.44919490321444497441013971689, 11.25963686993249342239542037875, 12.57861362048066741077805147256
