L(s) = 1 | + (0.173 + 0.984i)2-s + (1.17 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (1.17 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.233 − 1.32i)9-s + (0.939 + 1.62i)11-s + (−0.766 + 1.32i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 1.34·18-s + (−1.43 + 1.20i)22-s + (−1.43 − 0.524i)24-s + (0.766 + 0.642i)25-s + (−0.266 − 0.460i)27-s + ⋯ |
L(s) = 1 | + (0.173 + 0.984i)2-s + (1.17 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (1.17 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.233 − 1.32i)9-s + (0.939 + 1.62i)11-s + (−0.766 + 1.32i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 1.34·18-s + (−1.43 + 1.20i)22-s + (−1.43 − 0.524i)24-s + (0.766 + 0.642i)25-s + (−0.266 − 0.460i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.898368884\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.898368884\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
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
−8.911042730049055672830226354554, −8.104367119965576496477951046497, −7.31137452733484196859698453034, −7.05641943212562433989150434623, −6.37679197483382470330448627073, −5.15597203837535785034156849401, −4.38574192095224667266831788334, −3.45495230969651361557952023987, −2.46222792121493171702326681119, −1.35267881924331781436672777891,
1.27586851317820056133708197222, 2.60778618228833295019566931744, 3.22589506891299867878157489033, 3.98585870723683865255302232045, 4.45924373153766730608235898900, 5.65421471222022446086069859236, 6.36053211547273937746630940375, 7.84524197949401456520881581676, 8.609609104569426836445396083750, 8.899989715096337851940710935306