L(s) = 1 | + (−0.906 − 1.57i)5-s + 2.52·7-s − 0.813·11-s + (2.16 − 3.75i)13-s + (−0.458 − 0.793i)17-s + (−3.22 − 2.93i)19-s + (0.906 − 1.57i)23-s + (0.855 − 1.48i)25-s + (−4.00 + 6.94i)29-s + 9.30·31-s + (−2.28 − 3.96i)35-s − 11.3·37-s + (−5.60 − 9.70i)41-s + (4.98 + 8.63i)43-s + (2.82 − 4.88i)47-s + ⋯ |
L(s) = 1 | + (−0.405 − 0.702i)5-s + 0.954·7-s − 0.245·11-s + (0.601 − 1.04i)13-s + (−0.111 − 0.192i)17-s + (−0.738 − 0.673i)19-s + (0.189 − 0.327i)23-s + (0.171 − 0.296i)25-s + (−0.744 + 1.28i)29-s + 1.67·31-s + (−0.386 − 0.670i)35-s − 1.85·37-s + (−0.875 − 1.51i)41-s + (0.759 + 1.31i)43-s + (0.411 − 0.713i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.448940272\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.448940272\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.22 + 2.93i)T \) |
good | 5 | \( 1 + (0.906 + 1.57i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 + 0.813T + 11T^{2} \) |
| 13 | \( 1 + (-2.16 + 3.75i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.458 + 0.793i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.906 + 1.57i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.00 - 6.94i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.30T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + (5.60 + 9.70i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.98 - 8.63i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.82 + 4.88i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.45 + 2.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.948 + 1.64i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.29 - 5.71i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.57 + 2.72i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.355 - 0.615i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.313 + 0.543i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.16 + 8.95i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.86T + 83T^{2} \) |
| 89 | \( 1 + (-6.79 + 11.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.31 + 4.00i)T + (-48.5 + 84.0i)T^{2} \) |
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
−8.641838780877629360818469060570, −7.985785783240269692826895775880, −7.19319290313578463212745089698, −6.27380412467524778752507018268, −5.17680521936985781369616198696, −4.85623612332619466556055937417, −3.86433503015979073256997699848, −2.82609838140570486517082684665, −1.61646414625101143280528728787, −0.48055133231207046451476981210,
1.43782517515046950310239998352, 2.36435130033824793497555709406, 3.55835698384995504430722405664, 4.25338431021862868195134158749, 5.10845650414202668620187585076, 6.14527549968904043384674897903, 6.76931976262649834663033792081, 7.64013809059064853047785469026, 8.242365241711919321092896578726, 8.907550458693069027086590557364