| L(s) = 1 | − 8.54·3-s − 18.5·5-s − 7·7-s + 46.0·9-s + 63.2·11-s + 16.7·13-s + 158.·15-s − 41.2·17-s + 39.1·19-s + 59.8·21-s + 21.8·23-s + 219.·25-s − 163.·27-s − 138.·29-s + 95.6·31-s − 541.·33-s + 129.·35-s − 176.·37-s − 143.·39-s − 407.·41-s + 100.·43-s − 855.·45-s + 144.·47-s + 49·49-s + 353.·51-s + 409.·53-s − 1.17e3·55-s + ⋯ |
| L(s) = 1 | − 1.64·3-s − 1.65·5-s − 0.377·7-s + 1.70·9-s + 1.73·11-s + 0.357·13-s + 2.72·15-s − 0.589·17-s + 0.472·19-s + 0.621·21-s + 0.197·23-s + 1.75·25-s − 1.16·27-s − 0.884·29-s + 0.554·31-s − 2.85·33-s + 0.627·35-s − 0.784·37-s − 0.587·39-s − 1.55·41-s + 0.355·43-s − 2.83·45-s + 0.447·47-s + 0.142·49-s + 0.969·51-s + 1.06·53-s − 2.87·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 + 8.54T + 27T^{2} \) |
| 5 | \( 1 + 18.5T + 125T^{2} \) |
| 11 | \( 1 - 63.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 41.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 21.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 95.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 176.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 407.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 100.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 144.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 0.852T + 2.05e5T^{2} \) |
| 61 | \( 1 - 407.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 9.38T + 3.00e5T^{2} \) |
| 71 | \( 1 - 944.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 86.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 563.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 969.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 956.T + 9.12e5T^{2} \) |
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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
−10.63346796847240952539301297116, −9.401229476478307572103606603017, −8.357798263289113964455571349962, −7.00352959627244803103514659130, −6.68852562168868138858383206084, −5.45099377738345949590875005530, −4.29129134220235600695250641105, −3.64967147937997638905277860692, −1.10408659547445608114688022172, 0,
1.10408659547445608114688022172, 3.64967147937997638905277860692, 4.29129134220235600695250641105, 5.45099377738345949590875005530, 6.68852562168868138858383206084, 7.00352959627244803103514659130, 8.357798263289113964455571349962, 9.401229476478307572103606603017, 10.63346796847240952539301297116
