L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.707 − 0.707i)4-s + (−0.324 + 0.216i)5-s + (−0.382 + 0.923i)8-s + (−0.923 − 0.382i)9-s + (0.216 − 0.324i)10-s + (1 − i)13-s − i·16-s + (−0.382 − 0.923i)17-s + 18-s + (−0.0761 + 0.382i)20-s + (−0.324 + 0.783i)25-s + (−0.541 + 1.30i)26-s + (−0.324 + 0.216i)29-s + (0.382 + 0.923i)32-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.707 − 0.707i)4-s + (−0.324 + 0.216i)5-s + (−0.382 + 0.923i)8-s + (−0.923 − 0.382i)9-s + (0.216 − 0.324i)10-s + (1 − i)13-s − i·16-s + (−0.382 − 0.923i)17-s + 18-s + (−0.0761 + 0.382i)20-s + (−0.324 + 0.783i)25-s + (−0.541 + 1.30i)26-s + (−0.324 + 0.216i)29-s + (0.382 + 0.923i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3892487903\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3892487903\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.923 - 0.382i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.382 + 0.923i)T \) |
good | 3 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 5 | \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \) |
| 11 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (1.92 + 0.382i)T + (0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (-1.63 + 1.08i)T + (0.382 - 0.923i)T^{2} \) |
| 79 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \) |
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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
−8.572292833086110774760325267445, −8.000576382539912409772309872986, −7.13937235640256281879985743615, −6.56017600841740805959613273075, −5.61101831134088487925526632543, −5.18618738697365218445112034323, −3.60153553870245231910924115158, −2.98650488985074002478277665291, −1.74538984739385140886013182202, −0.31048825321344191085752977878,
1.45979617298016292532371462923, 2.32936877420768256629508959265, 3.46540383603642607143530869685, 4.09951675153899166528605124125, 5.27325265667360744883563711569, 6.35025976800784638779364639627, 6.74570975115924021033362949712, 7.913655935373203687037032343064, 8.430420650009190296292511368430, 8.799848373762707315049003848866