| L(s) = 1 | − 3·3-s − 5·5-s − 14.3·7-s + 9·9-s − 17.0·11-s − 13·13-s + 15·15-s − 15.6·17-s + 16.7·19-s + 43.0·21-s + 127.·23-s + 25·25-s − 27·27-s + 293.·29-s + 280.·31-s + 51.1·33-s + 71.7·35-s − 257.·37-s + 39·39-s − 39.0·41-s − 59.7·43-s − 45·45-s − 488.·47-s − 136.·49-s + 46.9·51-s − 235.·53-s + 85.2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.775·7-s + 0.333·9-s − 0.467·11-s − 0.277·13-s + 0.258·15-s − 0.223·17-s + 0.202·19-s + 0.447·21-s + 1.15·23-s + 0.200·25-s − 0.192·27-s + 1.87·29-s + 1.62·31-s + 0.269·33-s + 0.346·35-s − 1.14·37-s + 0.160·39-s − 0.148·41-s − 0.211·43-s − 0.149·45-s − 1.51·47-s − 0.398·49-s + 0.128·51-s − 0.609·53-s + 0.208·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| good | 7 | \( 1 + 14.3T + 343T^{2} \) |
| 11 | \( 1 + 17.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 15.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 127.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 280.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 39.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 59.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 235.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 469.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 24.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 649.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 286.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 377.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 67.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.50e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 990.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−8.592399819942551280535178769226, −7.905054528541668621163080597343, −6.73065889741204450604759331389, −6.54834592837938571333739430747, −5.20816860089348980165750430478, −4.67344099583152164895602241474, −3.45320671983816785270696795292, −2.63636051792285127485543720204, −1.04808214251369306798675517174, 0,
1.04808214251369306798675517174, 2.63636051792285127485543720204, 3.45320671983816785270696795292, 4.67344099583152164895602241474, 5.20816860089348980165750430478, 6.54834592837938571333739430747, 6.73065889741204450604759331389, 7.905054528541668621163080597343, 8.592399819942551280535178769226
