| L(s) = 1 | − 5.95·2-s + 9·3-s + 3.48·4-s − 53.6·6-s − 79.6·7-s + 169.·8-s + 81·9-s − 121·11-s + 31.3·12-s + 950.·13-s + 474.·14-s − 1.12e3·16-s + 383.·17-s − 482.·18-s − 523.·19-s − 717.·21-s + 720.·22-s − 4.47e3·23-s + 1.52e3·24-s − 5.66e3·26-s + 729·27-s − 277.·28-s + 2.76e3·29-s − 8.30e3·31-s + 1.25e3·32-s − 1.08e3·33-s − 2.28e3·34-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 0.577·3-s + 0.108·4-s − 0.607·6-s − 0.614·7-s + 0.938·8-s + 0.333·9-s − 0.301·11-s + 0.0628·12-s + 1.56·13-s + 0.647·14-s − 1.09·16-s + 0.321·17-s − 0.351·18-s − 0.332·19-s − 0.354·21-s + 0.317·22-s − 1.76·23-s + 0.541·24-s − 1.64·26-s + 0.192·27-s − 0.0669·28-s + 0.610·29-s − 1.55·31-s + 0.216·32-s − 0.174·33-s − 0.338·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 5.95T + 32T^{2} \) |
| 7 | \( 1 + 79.6T + 1.68e4T^{2} \) |
| 13 | \( 1 - 950.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 383.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 523.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.47e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.30e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.51e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.27e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.24e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.85e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.28e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.40e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.06e5T + 8.58e9T^{2} \) |
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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
−9.173060015637171273472551251601, −8.179227795503064962179820921288, −7.82041690706011847045104269406, −6.63149905243217125907566101657, −5.73979284954002402839180778156, −4.27870081562550163741955259694, −3.52247004354228386914134909311, −2.17436978639290198945616391443, −1.13194860685337119040362853930, 0,
1.13194860685337119040362853930, 2.17436978639290198945616391443, 3.52247004354228386914134909311, 4.27870081562550163741955259694, 5.73979284954002402839180778156, 6.63149905243217125907566101657, 7.82041690706011847045104269406, 8.179227795503064962179820921288, 9.173060015637171273472551251601
