| L(s) = 1 | − 3-s − 5-s − 2.20i·7-s + 9-s + 4.04i·11-s − 1.23i·13-s + 15-s − 4.40·17-s + (−3.61 + 2.44i)19-s + 2.20i·21-s − 9.21i·23-s + 25-s − 27-s + 8.22i·29-s + 7.30·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.833i·7-s + 0.333·9-s + 1.22i·11-s − 0.342i·13-s + 0.258·15-s − 1.06·17-s + (−0.828 + 0.559i)19-s + 0.481i·21-s − 1.92i·23-s + 0.200·25-s − 0.192·27-s + 1.52i·29-s + 1.31·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.051044301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.051044301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (3.61 - 2.44i)T \) |
| good | 7 | \( 1 + 2.20iT - 7T^{2} \) |
| 11 | \( 1 - 4.04iT - 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 23 | \( 1 + 9.21iT - 23T^{2} \) |
| 29 | \( 1 - 8.22iT - 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 - 3.55iT - 37T^{2} \) |
| 41 | \( 1 - 4.31iT - 41T^{2} \) |
| 43 | \( 1 - 7.01iT - 43T^{2} \) |
| 47 | \( 1 + 10.4iT - 47T^{2} \) |
| 53 | \( 1 + 7.55iT - 53T^{2} \) |
| 59 | \( 1 - 0.470T + 59T^{2} \) |
| 61 | \( 1 - 0.970T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 0.320T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 9.32iT - 83T^{2} \) |
| 89 | \( 1 - 4.17iT - 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−8.281883207090249833308584691736, −7.42925993095449690501641239413, −6.66343093575636203948049278855, −6.46103234460000335609944629288, −5.08653918200163223191219718831, −4.51567484403401318916345481225, −4.03415414236093148585032371740, −2.81458320114223163695292264571, −1.74954120432715686312855566126, −0.52097185661878498723554959222,
0.64684328130417903375911065859, 2.03917942872216110985821589993, 2.93062487626992969885037561994, 3.98857900951211523395228607275, 4.60528892331317178562121998002, 5.70981585445620608233608024303, 5.98005333673111165578729665095, 6.86682905929472377883131172457, 7.63821099506964482069090107504, 8.506859905118654764344335370228
