| L(s) = 1 | − 3.87·2-s + 3·3-s + 7.00·4-s − 5·5-s − 11.6·6-s + 15.4·7-s + 3.87·8-s + 9·9-s + 19.3·10-s + 21.0·12-s − 69.7·13-s − 60.0·14-s − 15·15-s − 70.9·16-s + 15.4·17-s − 34.8·18-s − 7.74·19-s − 35.0·20-s + 46.4·21-s + 48·23-s + 11.6·24-s + 25·25-s + 270·26-s + 27·27-s + 108.·28-s + 209.·29-s + 58.0·30-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.577·3-s + 0.875·4-s − 0.447·5-s − 0.790·6-s + 0.836·7-s + 0.171·8-s + 0.333·9-s + 0.612·10-s + 0.505·12-s − 1.48·13-s − 1.14·14-s − 0.258·15-s − 1.10·16-s + 0.221·17-s − 0.456·18-s − 0.0935·19-s − 0.391·20-s + 0.482·21-s + 0.435·23-s + 0.0988·24-s + 0.200·25-s + 2.03·26-s + 0.192·27-s + 0.731·28-s + 1.33·29-s + 0.353·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 3.87T + 8T^{2} \) |
| 7 | \( 1 - 15.4T + 343T^{2} \) |
| 13 | \( 1 + 69.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 15.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 7.74T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48T + 1.21e4T^{2} \) |
| 29 | \( 1 - 209.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 160T + 2.97e4T^{2} \) |
| 37 | \( 1 + 266T + 5.06e4T^{2} \) |
| 41 | \( 1 + 178.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 309.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 504T + 1.03e5T^{2} \) |
| 53 | \( 1 + 342T + 1.48e5T^{2} \) |
| 59 | \( 1 + 660T + 2.05e5T^{2} \) |
| 61 | \( 1 - 216.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 496T + 3.00e5T^{2} \) |
| 71 | \( 1 + 708T + 3.57e5T^{2} \) |
| 73 | \( 1 - 642.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 178.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 85.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 606T + 7.04e5T^{2} \) |
| 97 | \( 1 - 254T + 9.12e5T^{2} \) |
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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
−8.377101091443660710738583181545, −8.015174743625218510657658139800, −7.30715583388747158873968004724, −6.60918771942042557881963038122, −4.96316912492890356884526983238, −4.54334867391187319414925652661, −3.11281506120626705124150229800, −2.12867523949212963342903612660, −1.16494356637207270430693701809, 0,
1.16494356637207270430693701809, 2.12867523949212963342903612660, 3.11281506120626705124150229800, 4.54334867391187319414925652661, 4.96316912492890356884526983238, 6.60918771942042557881963038122, 7.30715583388747158873968004724, 8.015174743625218510657658139800, 8.377101091443660710738583181545
