| L(s) = 1 | − 7.15·3-s − 9.75·5-s + 7·7-s + 24.2·9-s − 11·11-s + 1.08·13-s + 69.8·15-s − 95.3·17-s + 77.1·19-s − 50.1·21-s − 114.·23-s − 29.8·25-s + 19.5·27-s + 280.·29-s + 233.·31-s + 78.7·33-s − 68.2·35-s − 294.·37-s − 7.76·39-s − 136.·41-s + 312.·43-s − 236.·45-s + 476.·47-s + 49·49-s + 682.·51-s + 283.·53-s + 107.·55-s + ⋯ |
| L(s) = 1 | − 1.37·3-s − 0.872·5-s + 0.377·7-s + 0.898·9-s − 0.301·11-s + 0.0231·13-s + 1.20·15-s − 1.36·17-s + 0.931·19-s − 0.520·21-s − 1.03·23-s − 0.238·25-s + 0.139·27-s + 1.79·29-s + 1.35·31-s + 0.415·33-s − 0.329·35-s − 1.31·37-s − 0.0318·39-s − 0.520·41-s + 1.10·43-s − 0.784·45-s + 1.47·47-s + 0.142·49-s + 1.87·51-s + 0.733·53-s + 0.263·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 7.15T + 27T^{2} \) |
| 5 | \( 1 + 9.75T + 125T^{2} \) |
| 13 | \( 1 - 1.08T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 280.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 233.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 312.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 476.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 706.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 25.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 869.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 197.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 821.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 668.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 542.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.70e3T + 9.12e5T^{2} \) |
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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
−8.793713719820650206828165670276, −8.075754075785216482934358002605, −7.14403616771503572068666187583, −6.41564424027374552298800437067, −5.51079988182363948492373920307, −4.69463853665594786460226571585, −3.99212615176713428861092334138, −2.53404367153357959049816068846, −0.971984813573814089143827776761, 0,
0.971984813573814089143827776761, 2.53404367153357959049816068846, 3.99212615176713428861092334138, 4.69463853665594786460226571585, 5.51079988182363948492373920307, 6.41564424027374552298800437067, 7.14403616771503572068666187583, 8.075754075785216482934358002605, 8.793713719820650206828165670276
