L(s) = 1 | + (0.989 − 0.142i)2-s + (−0.540 + 0.841i)3-s + (0.959 − 0.281i)4-s + (0.654 − 0.755i)5-s + (−0.415 + 0.909i)6-s + (−0.909 − 1.41i)7-s + (0.909 − 0.415i)8-s + (−0.415 − 0.909i)9-s + (0.540 − 0.841i)10-s + (−0.281 + 0.959i)12-s + (−1.10 − 1.27i)14-s + (0.281 + 0.959i)15-s + (0.841 − 0.540i)16-s + (−0.540 − 0.841i)18-s + (0.415 − 0.909i)20-s + 1.68·21-s + ⋯ |
L(s) = 1 | + (0.989 − 0.142i)2-s + (−0.540 + 0.841i)3-s + (0.959 − 0.281i)4-s + (0.654 − 0.755i)5-s + (−0.415 + 0.909i)6-s + (−0.909 − 1.41i)7-s + (0.909 − 0.415i)8-s + (−0.415 − 0.909i)9-s + (0.540 − 0.841i)10-s + (−0.281 + 0.959i)12-s + (−1.10 − 1.27i)14-s + (0.281 + 0.959i)15-s + (0.841 − 0.540i)16-s + (−0.540 − 0.841i)18-s + (0.415 − 0.909i)20-s + 1.68·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.693820330\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.693820330\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.989 + 0.142i)T \) |
| 3 | \( 1 + (0.540 - 0.841i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (0.989 - 0.142i)T \) |
good | 7 | \( 1 + (0.909 + 1.41i)T + (-0.415 + 0.909i)T^{2} \) |
| 11 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 17 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 19 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 29 | \( 1 + (0.425 - 1.45i)T + (-0.841 - 0.540i)T^{2} \) |
| 31 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 37 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 41 | \( 1 + (-0.817 - 0.708i)T + (0.142 + 0.989i)T^{2} \) |
| 43 | \( 1 + (-1.74 - 0.797i)T + (0.654 + 0.755i)T^{2} \) |
| 47 | \( 1 - 1.30iT - T^{2} \) |
| 53 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 61 | \( 1 + (0.512 - 0.234i)T + (0.654 - 0.755i)T^{2} \) |
| 67 | \( 1 + (-0.822 + 0.118i)T + (0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 73 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 83 | \( 1 + (-1.19 - 1.37i)T + (-0.142 + 0.989i)T^{2} \) |
| 89 | \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 97 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
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
−9.705731865985118296492173090377, −9.417746310723507121640660181784, −7.921265633173985402685536979781, −6.86623136919167340454235078576, −6.15224370279315433771349547853, −5.45372441908516765514782539946, −4.44509989215580660412211149407, −3.96798830224303967803399838359, −2.91634346084654105258023006857, −1.17396967733984081395037071282,
2.17894150041089117732224727886, 2.50239087832729079276722721331, 3.75734079326222341122972682451, 5.25168417885462620150083659524, 5.96469339209117517706733553030, 6.20911828191601666583584853706, 7.10915883698733225528153796068, 7.927510277573419591479365351345, 9.057498602919680957691477057663, 10.04623237271326596525037291813