| L(s) = 1 | − 2.34i·3-s − 1.19i·7-s − 2.48·9-s + 4.97·11-s − 6.63i·13-s − 1.48i·17-s − 19-s − 2.80·21-s − 0.510i·23-s − 1.19i·27-s + 7.88·29-s − 2.97·31-s − 11.6i·33-s + 7.14i·37-s − 15.5·39-s + ⋯ |
| L(s) = 1 | − 1.35i·3-s − 0.452i·7-s − 0.829·9-s + 1.50·11-s − 1.84i·13-s − 0.361i·17-s − 0.229·19-s − 0.611·21-s − 0.106i·23-s − 0.230i·27-s + 1.46·29-s − 0.534·31-s − 2.03i·33-s + 1.17i·37-s − 2.48·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.026733684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026733684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.34iT - 3T^{2} \) |
| 7 | \( 1 + 1.19iT - 7T^{2} \) |
| 11 | \( 1 - 4.97T + 11T^{2} \) |
| 13 | \( 1 + 6.63iT - 13T^{2} \) |
| 17 | \( 1 + 1.48iT - 17T^{2} \) |
| 23 | \( 1 + 0.510iT - 23T^{2} \) |
| 29 | \( 1 - 7.88T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 - 7.14iT - 37T^{2} \) |
| 41 | \( 1 - 1.66T + 41T^{2} \) |
| 43 | \( 1 + 6.39iT - 43T^{2} \) |
| 47 | \( 1 - 9.95iT - 47T^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 3.66T + 61T^{2} \) |
| 67 | \( 1 + 7.61iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 8.68iT - 83T^{2} \) |
| 89 | \( 1 - 4.87T + 89T^{2} \) |
| 97 | \( 1 - 6.81iT - 97T^{2} \) |
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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
−8.202334079826398577968530576789, −7.30790813403473850467993282873, −6.82283956249643727528146916485, −6.18166190694317414208385170469, −5.36999996758922189384965001058, −4.30813300771414593835423957170, −3.35688173548907410371942501612, −2.46380480121972654620512450713, −1.28158527650333582157824441116, −0.67022896519519713265701968297,
1.42479544875468443842243438656, 2.50850159937470630845207701855, 3.84189747281471888938144246313, 4.05963809433703045354104343816, 4.83092254224411634793128115011, 5.78854182948266722527228598940, 6.55690714016223374589213878630, 7.16453019012130466543084254740, 8.475673643458204871867374005814, 9.086879580592584972586523372325
