| L(s) = 1 | − 2.52·3-s − 3.46·5-s + 1.96·7-s + 3.36·9-s + 1.48·11-s + 5.95·13-s + 8.75·15-s + 17-s − 3.32·19-s − 4.95·21-s + 6.67·23-s + 7.03·25-s − 0.930·27-s + 4.76·29-s − 0.104·31-s − 3.74·33-s − 6.80·35-s − 10.5·37-s − 15.0·39-s − 5.65·41-s − 10.0·43-s − 11.6·45-s + 6.74·47-s − 3.14·49-s − 2.52·51-s − 3.21·53-s − 5.15·55-s + ⋯ |
| L(s) = 1 | − 1.45·3-s − 1.55·5-s + 0.741·7-s + 1.12·9-s + 0.447·11-s + 1.65·13-s + 2.26·15-s + 0.242·17-s − 0.763·19-s − 1.08·21-s + 1.39·23-s + 1.40·25-s − 0.178·27-s + 0.884·29-s − 0.0188·31-s − 0.652·33-s − 1.15·35-s − 1.73·37-s − 2.40·39-s − 0.882·41-s − 1.53·43-s − 1.74·45-s + 0.983·47-s − 0.449·49-s − 0.353·51-s − 0.441·53-s − 0.694·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8711123819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8711123819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 23 | \( 1 - 6.67T + 23T^{2} \) |
| 29 | \( 1 - 4.76T + 29T^{2} \) |
| 31 | \( 1 + 0.104T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 2.55T + 73T^{2} \) |
| 79 | \( 1 - 7.41T + 79T^{2} \) |
| 83 | \( 1 + 4.50T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 4.10T + 97T^{2} \) |
| 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
−8.556769263771704268764609113714, −7.67277074315363203309298077943, −6.72959483415546802509985166445, −6.45744974129844245799707618280, −5.27652308796756401265901412513, −4.82473075717790185294085950490, −3.94991276173582948126692413857, −3.32783963686964052063833722661, −1.50055654509949925966650621358, −0.62940731895209176535047522383,
0.62940731895209176535047522383, 1.50055654509949925966650621358, 3.32783963686964052063833722661, 3.94991276173582948126692413857, 4.82473075717790185294085950490, 5.27652308796756401265901412513, 6.45744974129844245799707618280, 6.72959483415546802509985166445, 7.67277074315363203309298077943, 8.556769263771704268764609113714
