| L(s) = 1 | − 0.705·2-s − 3-s − 1.50·4-s + 5-s + 0.705·6-s − 2.52·7-s + 2.46·8-s + 9-s − 0.705·10-s − 2.77·11-s + 1.50·12-s + 6.65·13-s + 1.77·14-s − 15-s + 1.26·16-s − 3.40·17-s − 0.705·18-s + 4.59·19-s − 1.50·20-s + 2.52·21-s + 1.95·22-s − 2.73·23-s − 2.46·24-s + 25-s − 4.68·26-s − 27-s + 3.78·28-s + ⋯ |
| L(s) = 1 | − 0.498·2-s − 0.577·3-s − 0.751·4-s + 0.447·5-s + 0.287·6-s − 0.952·7-s + 0.873·8-s + 0.333·9-s − 0.222·10-s − 0.837·11-s + 0.433·12-s + 1.84·13-s + 0.474·14-s − 0.258·15-s + 0.316·16-s − 0.826·17-s − 0.166·18-s + 1.05·19-s − 0.336·20-s + 0.549·21-s + 0.417·22-s − 0.570·23-s − 0.504·24-s + 0.200·25-s − 0.919·26-s − 0.192·27-s + 0.715·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.705T + 2T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + 2.77T + 11T^{2} \) |
| 13 | \( 1 - 6.65T + 13T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 - 4.59T + 19T^{2} \) |
| 23 | \( 1 + 2.73T + 23T^{2} \) |
| 29 | \( 1 + 7.54T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 - 8.84T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 9.96T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 1.94T + 73T^{2} \) |
| 79 | \( 1 - 8.73T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 - 7.69T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.927564243183735540604128716890, −6.85773034447773542465464816525, −6.40169390646927885628475267614, −5.42795028660055166185024517164, −5.16145058357431290569651586563, −3.86029695827241067310317128065, −3.49345456258864748549031844086, −2.09645590961256714349195772518, −1.03564600595234065227293741946, 0,
1.03564600595234065227293741946, 2.09645590961256714349195772518, 3.49345456258864748549031844086, 3.86029695827241067310317128065, 5.16145058357431290569651586563, 5.42795028660055166185024517164, 6.40169390646927885628475267614, 6.85773034447773542465464816525, 7.927564243183735540604128716890
