L(s) = 1 | − 2.16i·2-s − i·3-s − 2.66·4-s − 2.16·6-s + 3.16i·7-s + 1.44i·8-s − 9-s + 1.53·11-s + 2.66i·12-s − 5.24i·13-s + 6.82·14-s − 2.21·16-s − 1.29i·17-s + 2.16i·18-s − 5.44·19-s + ⋯ |
L(s) = 1 | − 1.52i·2-s − 0.577i·3-s − 1.33·4-s − 0.882·6-s + 1.19i·7-s + 0.510i·8-s − 0.333·9-s + 0.463·11-s + 0.770i·12-s − 1.45i·13-s + 1.82·14-s − 0.554·16-s − 0.313i·17-s + 0.509i·18-s − 1.24·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6685609857\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6685609857\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2.16iT - 2T^{2} \) |
| 7 | \( 1 - 3.16iT - 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 + 5.24iT - 13T^{2} \) |
| 17 | \( 1 + 1.29iT - 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + 6.44iT - 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 - 5.95iT - 37T^{2} \) |
| 41 | \( 1 + 8.53T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 0.753iT - 47T^{2} \) |
| 53 | \( 1 - 9.74iT - 53T^{2} \) |
| 59 | \( 1 + 4.11T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.89iT - 67T^{2} \) |
| 71 | \( 1 + 0.0708T + 71T^{2} \) |
| 73 | \( 1 - 4.01iT - 73T^{2} \) |
| 79 | \( 1 + 1.61T + 79T^{2} \) |
| 83 | \( 1 - 13.1iT - 83T^{2} \) |
| 89 | \( 1 + 7.27T + 89T^{2} \) |
| 97 | \( 1 + 10.1iT - 97T^{2} \) |
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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
−8.647536775388992765886924171240, −8.352790568032772055921905948674, −7.01296934676291805367572513693, −6.16240979805112758311871299717, −5.27354927401308012998611398408, −4.26934441452966839564873081881, −3.08040331711345721272655114842, −2.54087177068781203711627535601, −1.57868166108979313782696199611, −0.23641223569159612741831971083,
1.78458290345125681353377227069, 3.64859303888121080644230898845, 4.28692178880478053389908343981, 4.96825053520057456379830711449, 6.08758383946314361456323788683, 6.66226129632403016715546358892, 7.33148687051331885805943030540, 8.086845016059991721923508246001, 8.986634128795067727522309784456, 9.438202993611743248644448800938