L(s) = 1 | + (0.723 − 0.690i)2-s + (0.0930 − 0.647i)3-s + (0.0475 − 0.998i)4-s + (−0.379 − 0.532i)6-s + (−0.654 − 0.755i)8-s + (0.548 + 0.161i)9-s + (−0.759 + 1.06i)11-s + (−0.642 − 0.123i)12-s + (−0.995 − 0.0950i)16-s + (−0.0845 − 1.77i)17-s + (0.508 − 0.262i)18-s + (−0.469 + 1.93i)19-s + (0.186 + 1.29i)22-s + (−0.550 + 0.353i)24-s + (−0.654 + 0.755i)25-s + ⋯ |
L(s) = 1 | + (0.723 − 0.690i)2-s + (0.0930 − 0.647i)3-s + (0.0475 − 0.998i)4-s + (−0.379 − 0.532i)6-s + (−0.654 − 0.755i)8-s + (0.548 + 0.161i)9-s + (−0.759 + 1.06i)11-s + (−0.642 − 0.123i)12-s + (−0.995 − 0.0950i)16-s + (−0.0845 − 1.77i)17-s + (0.508 − 0.262i)18-s + (−0.469 + 1.93i)19-s + (0.186 + 1.29i)22-s + (−0.550 + 0.353i)24-s + (−0.654 + 0.755i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.278722415\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.278722415\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.723 + 0.690i)T \) |
| 67 | \( 1 + (-0.415 - 0.909i)T \) |
good | 3 | \( 1 + (-0.0930 + 0.647i)T + (-0.959 - 0.281i)T^{2} \) |
| 5 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 7 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 11 | \( 1 + (0.759 - 1.06i)T + (-0.327 - 0.945i)T^{2} \) |
| 13 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 17 | \( 1 + (0.0845 + 1.77i)T + (-0.995 + 0.0950i)T^{2} \) |
| 19 | \( 1 + (0.469 - 1.93i)T + (-0.888 - 0.458i)T^{2} \) |
| 23 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.70 - 0.879i)T + (0.580 + 0.814i)T^{2} \) |
| 43 | \( 1 + (-0.975 + 0.627i)T + (0.415 - 0.909i)T^{2} \) |
| 47 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 53 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 59 | \( 1 + (1.28 + 1.48i)T + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 71 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 73 | \( 1 + (-0.0552 - 0.0775i)T + (-0.327 + 0.945i)T^{2} \) |
| 79 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 83 | \( 1 + (0.827 + 0.0789i)T + (0.981 + 0.189i)T^{2} \) |
| 89 | \( 1 + (0.205 + 1.43i)T + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.92798869869932711337707468526, −9.974424988901836271190250034875, −9.465613368686473030114008830684, −7.84505322249012947733096316904, −7.21806750917104312853787945435, −6.08806245356202248290286319653, −5.02865963947302203505757943598, −4.10755703309075446505343039832, −2.66592575198728962847384038119, −1.64022771488741472356352291560,
2.65059047860534385449777820188, 3.86307405577615865936142914033, 4.59911520141905224933369458715, 5.73760427596897279897482345193, 6.53062422800262724986476219596, 7.67041950023874865248879903800, 8.525089563652020350103469186846, 9.309137987663014876376722320508, 10.64519210921574197212182939896, 11.08621978435671663569381997418