| L(s) = 1 | + 9.26·2-s − 9·3-s + 53.9·4-s + 17.8·5-s − 83.4·6-s + 23.9·7-s + 202.·8-s + 81·9-s + 164.·10-s − 507.·11-s − 485.·12-s + 213.·13-s + 221.·14-s − 160.·15-s + 156.·16-s − 1.26e3·17-s + 750.·18-s − 1.82e3·19-s + 959.·20-s − 215.·21-s − 4.70e3·22-s + 2.30e3·23-s − 1.82e3·24-s − 2.80e3·25-s + 1.97e3·26-s − 729·27-s + 1.29e3·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 0.577·3-s + 1.68·4-s + 0.318·5-s − 0.945·6-s + 0.184·7-s + 1.12·8-s + 0.333·9-s + 0.521·10-s − 1.26·11-s − 0.972·12-s + 0.349·13-s + 0.302·14-s − 0.183·15-s + 0.152·16-s − 1.06·17-s + 0.546·18-s − 1.16·19-s + 0.536·20-s − 0.106·21-s − 2.07·22-s + 0.909·23-s − 0.647·24-s − 0.898·25-s + 0.572·26-s − 0.192·27-s + 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 - 9.26T + 32T^{2} \) |
| 5 | \( 1 - 17.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 23.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 507.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 213.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.26e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.82e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.09e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.70T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.25e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.80e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.27e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−10.20644947678630347269349096207, −8.828324841256524714575773063730, −7.58212286339600155097181931473, −6.49400264308041593990216460336, −5.85536650221438632643331296811, −4.91857842796717238663092983681, −4.25169755446414586930248361499, −2.90356250903744187612325890097, −1.90743392962276130709078292113, 0,
1.90743392962276130709078292113, 2.90356250903744187612325890097, 4.25169755446414586930248361499, 4.91857842796717238663092983681, 5.85536650221438632643331296811, 6.49400264308041593990216460336, 7.58212286339600155097181931473, 8.828324841256524714575773063730, 10.20644947678630347269349096207
