| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 8.86·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 17.7·10-s − 7.07·11-s − 12·12-s − 31.1·13-s + 14·14-s + 26.5·15-s + 16·16-s + 95.0·17-s − 18·18-s − 14.6·19-s − 35.4·20-s + 21·21-s + 14.1·22-s + 23·23-s + 24·24-s − 46.4·25-s + 62.3·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.792·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.560·10-s − 0.193·11-s − 0.288·12-s − 0.665·13-s + 0.267·14-s + 0.457·15-s + 0.250·16-s + 1.35·17-s − 0.235·18-s − 0.176·19-s − 0.396·20-s + 0.218·21-s + 0.137·22-s + 0.208·23-s + 0.204·24-s − 0.371·25-s + 0.470·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 8.86T + 125T^{2} \) |
| 11 | \( 1 + 7.07T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.6T + 6.85e3T^{2} \) |
| 29 | \( 1 + 1.23T + 2.43e4T^{2} \) |
| 31 | \( 1 - 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 302.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 51.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 109.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 89.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 616.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 575.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 215.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 652.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 780.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 422.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 677.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 886.T + 9.12e5T^{2} \) |
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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
−9.465418121070433847567782909144, −8.133577999716049528349280227863, −7.72063905343089221027273671015, −6.78610802612844182818151623672, −5.90521880418279776671311244070, −4.87227629895782597939501824455, −3.73793516851192387010246118006, −2.62261744342637951882252040247, −1.05236309864994985063539958768, 0,
1.05236309864994985063539958768, 2.62261744342637951882252040247, 3.73793516851192387010246118006, 4.87227629895782597939501824455, 5.90521880418279776671311244070, 6.78610802612844182818151623672, 7.72063905343089221027273671015, 8.133577999716049528349280227863, 9.465418121070433847567782909144
