| L(s) = 1 | + 9.76·2-s + 13.1·3-s + 63.4·4-s + 25·5-s + 128.·6-s − 183.·7-s + 307.·8-s − 69.0·9-s + 244.·10-s + 121·11-s + 836.·12-s − 1.05e3·13-s − 1.79e3·14-s + 329.·15-s + 971.·16-s + 1.31e3·17-s − 674.·18-s − 361·19-s + 1.58e3·20-s − 2.42e3·21-s + 1.18e3·22-s − 933.·23-s + 4.05e3·24-s + 625·25-s − 1.03e4·26-s − 4.11e3·27-s − 1.16e4·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 0.846·3-s + 1.98·4-s + 0.447·5-s + 1.46·6-s − 1.41·7-s + 1.69·8-s − 0.284·9-s + 0.772·10-s + 0.301·11-s + 1.67·12-s − 1.73·13-s − 2.44·14-s + 0.378·15-s + 0.948·16-s + 1.10·17-s − 0.490·18-s − 0.229·19-s + 0.886·20-s − 1.19·21-s + 0.520·22-s − 0.367·23-s + 1.43·24-s + 0.200·25-s − 2.99·26-s − 1.08·27-s − 2.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 9.76T + 32T^{2} \) |
| 3 | \( 1 - 13.1T + 243T^{2} \) |
| 7 | \( 1 + 183.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.31e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 933.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.29e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.33e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.67e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.18e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.31e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.40e4T + 8.58e9T^{2} \) |
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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
−8.869471837533614017241731572460, −7.56747495144479031197406687481, −6.90014783341887960606926680944, −5.96290013982269411641387492706, −5.37522728258712977891416110039, −4.23728188737607614432544201986, −3.28879624377910779473134328872, −2.82725516316632600293109612006, −1.95782653981947694975152867942, 0,
1.95782653981947694975152867942, 2.82725516316632600293109612006, 3.28879624377910779473134328872, 4.23728188737607614432544201986, 5.37522728258712977891416110039, 5.96290013982269411641387492706, 6.90014783341887960606926680944, 7.56747495144479031197406687481, 8.869471837533614017241731572460
