| L(s) = 1 | + 0.741·2-s − 1.44·4-s + (−2.05 + 0.880i)5-s + (−1.24 − 2.33i)7-s − 2.55·8-s + (−1.52 + 0.653i)10-s − 1.41i·11-s − 5.54·13-s + (−0.923 − 1.73i)14-s + 1.00·16-s + 6.07i·17-s − 7.12i·19-s + (2.97 − 1.27i)20-s − 1.04i·22-s − 4.78·23-s + ⋯ |
| L(s) = 1 | + 0.524·2-s − 0.724·4-s + (−0.919 + 0.393i)5-s + (−0.470 − 0.882i)7-s − 0.904·8-s + (−0.482 + 0.206i)10-s − 0.426i·11-s − 1.53·13-s + (−0.246 − 0.462i)14-s + 0.250·16-s + 1.47i·17-s − 1.63i·19-s + (0.666 − 0.285i)20-s − 0.223i·22-s − 0.997·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0262899 - 0.187058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0262899 - 0.187058i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.05 - 0.880i)T \) |
| 7 | \( 1 + (1.24 + 2.33i)T \) |
| good | 2 | \( 1 - 0.741T + 2T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 - 6.07iT - 17T^{2} \) |
| 19 | \( 1 + 7.12iT - 19T^{2} \) |
| 23 | \( 1 + 4.78T + 23T^{2} \) |
| 29 | \( 1 - 5.51iT - 29T^{2} \) |
| 31 | \( 1 - 1.30iT - 31T^{2} \) |
| 37 | \( 1 + 2.57iT - 37T^{2} \) |
| 41 | \( 1 - 5.95T + 41T^{2} \) |
| 43 | \( 1 - 6.76iT - 43T^{2} \) |
| 47 | \( 1 + 7.83iT - 47T^{2} \) |
| 53 | \( 1 + 9.90T + 53T^{2} \) |
| 59 | \( 1 + 1.84T + 59T^{2} \) |
| 61 | \( 1 + 11.6iT - 61T^{2} \) |
| 67 | \( 1 - 7.23iT - 67T^{2} \) |
| 71 | \( 1 + 8.34iT - 71T^{2} \) |
| 73 | \( 1 - 0.559T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 7.83iT - 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
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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
−11.27444525771299106172666041088, −10.37918891854339886215234370646, −9.416271954544754199529394109128, −8.299774070455012481503927394740, −7.34412903547229385995547140025, −6.34166109692024180313141711158, −4.87543431549971819373910235681, −4.03916813275047313694778874449, −3.03897931150445592527912206871, −0.11247118197453507645589789594,
2.72447476531906859358312338592, 4.05810540101675420063164783247, 4.95536521202198202161087663935, 5.91069451420716870949347093799, 7.42470508349129820166200001729, 8.263717445816835276032888079870, 9.439553211400995876621920188243, 9.865538872768520768495082538893, 11.67168104814767460949153249023, 12.26322572924736373977574461272
