L(s) = 1 | + (0.125 − 0.0683i)3-s + (−0.529 + 0.824i)9-s + (0.755 − 1.65i)11-s + (−0.281 − 0.0405i)17-s + (−0.398 − 1.83i)19-s + (0.654 − 0.755i)25-s + (−0.0201 + 0.281i)27-s + (−0.0185 − 0.258i)33-s + (0.677 − 1.24i)41-s + (0.559 + 1.50i)43-s + (0.755 + 0.654i)49-s + (−0.0380 + 0.0141i)51-s + (−0.175 − 0.202i)57-s + (1.05 + 0.574i)59-s + (−1.03 − 0.304i)67-s + ⋯ |
L(s) = 1 | + (0.125 − 0.0683i)3-s + (−0.529 + 0.824i)9-s + (0.755 − 1.65i)11-s + (−0.281 − 0.0405i)17-s + (−0.398 − 1.83i)19-s + (0.654 − 0.755i)25-s + (−0.0201 + 0.281i)27-s + (−0.0185 − 0.258i)33-s + (0.677 − 1.24i)41-s + (0.559 + 1.50i)43-s + (0.755 + 0.654i)49-s + (−0.0380 + 0.0141i)51-s + (−0.175 − 0.202i)57-s + (1.05 + 0.574i)59-s + (−1.03 − 0.304i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.220402500\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.220402500\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
good | 3 | \( 1 + (-0.125 + 0.0683i)T + (0.540 - 0.841i)T^{2} \) |
| 5 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 7 | \( 1 + (-0.755 - 0.654i)T^{2} \) |
| 11 | \( 1 + (-0.755 + 1.65i)T + (-0.654 - 0.755i)T^{2} \) |
| 13 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 17 | \( 1 + (0.281 + 0.0405i)T + (0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (0.398 + 1.83i)T + (-0.909 + 0.415i)T^{2} \) |
| 23 | \( 1 + (-0.909 + 0.415i)T^{2} \) |
| 29 | \( 1 + (0.755 + 0.654i)T^{2} \) |
| 31 | \( 1 + (-0.909 - 0.415i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.677 + 1.24i)T + (-0.540 - 0.841i)T^{2} \) |
| 43 | \( 1 + (-0.559 - 1.50i)T + (-0.755 + 0.654i)T^{2} \) |
| 47 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 53 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (-1.05 - 0.574i)T + (0.540 + 0.841i)T^{2} \) |
| 61 | \( 1 + (-0.989 - 0.142i)T^{2} \) |
| 67 | \( 1 + (1.03 + 0.304i)T + (0.841 + 0.540i)T^{2} \) |
| 71 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (-1.27 + 0.817i)T + (0.415 - 0.909i)T^{2} \) |
| 79 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 83 | \( 1 + (1.05 + 1.40i)T + (-0.281 + 0.959i)T^{2} \) |
| 97 | \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.866605893313333016607193844534, −8.267729515683853386120542257506, −7.38451382889827242323293517261, −6.51063196725391876898327926521, −5.88402236423719804607664215917, −4.97645569138379863221731359609, −4.15369034826797928362831374417, −3.03694883662048760761816664071, −2.39220257628288744897424258708, −0.822203507543702557402628435883,
1.41469767536455374211800885159, 2.41776056518296880540702149586, 3.67055981628687678678140515661, 4.16118667293124393516745735628, 5.22718365045411067010126462881, 6.09973667391901481632832625613, 6.81389122392317672280775277990, 7.49183881670673726383668863546, 8.431813240298046572020935581616, 9.085767611940258477409791960958