| L(s) = 1 | + 3.54·2-s + 4.59·4-s + 5·5-s − 21.2·7-s − 12.0·8-s + 17.7·10-s + 40.9·11-s − 8.89·13-s − 75.4·14-s − 79.6·16-s + 7.68·17-s + 40.7·19-s + 22.9·20-s + 145.·22-s + 80.7·23-s + 25·25-s − 31.5·26-s − 97.6·28-s − 29·29-s + 82.1·31-s − 185.·32-s + 27.2·34-s − 106.·35-s + 223.·37-s + 144.·38-s − 60.4·40-s + 274.·41-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.574·4-s + 0.447·5-s − 1.14·7-s − 0.534·8-s + 0.561·10-s + 1.12·11-s − 0.189·13-s − 1.44·14-s − 1.24·16-s + 0.109·17-s + 0.492·19-s + 0.256·20-s + 1.40·22-s + 0.731·23-s + 0.200·25-s − 0.238·26-s − 0.659·28-s − 0.185·29-s + 0.475·31-s − 1.02·32-s + 0.137·34-s − 0.513·35-s + 0.992·37-s + 0.617·38-s − 0.238·40-s + 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.858548112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.858548112\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 3.54T + 8T^{2} \) |
| 7 | \( 1 + 21.2T + 343T^{2} \) |
| 11 | \( 1 - 40.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.89T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.68T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 80.7T + 1.21e4T^{2} \) |
| 31 | \( 1 - 82.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 53.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 17.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 23.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 388.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 399.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 92.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 979.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 62.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 163.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
| 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
−9.464544056301145447514323150700, −8.645757908813601365661513795441, −7.23940883615385833476897027727, −6.47865931831363076916321176462, −5.94764447542265941063494401787, −5.03144659983891614715506777157, −4.05559119200299172787473409784, −3.31521218844110823993166677023, −2.43289973869564801422534187091, −0.819810116994125049973339714016,
0.819810116994125049973339714016, 2.43289973869564801422534187091, 3.31521218844110823993166677023, 4.05559119200299172787473409784, 5.03144659983891614715506777157, 5.94764447542265941063494401787, 6.47865931831363076916321176462, 7.23940883615385833476897027727, 8.645757908813601365661513795441, 9.464544056301145447514323150700
