| L(s) = 1 | − 2.61·2-s + 0.252·3-s + 4.85·4-s + 3.30·5-s − 0.660·6-s − 1.70·7-s − 7.47·8-s − 2.93·9-s − 8.65·10-s + 5.06·11-s + 1.22·12-s + 5.40·13-s + 4.47·14-s + 0.833·15-s + 9.86·16-s − 0.784·17-s + 7.68·18-s + 1.17·19-s + 16.0·20-s − 0.430·21-s − 13.2·22-s − 1.79·23-s − 1.88·24-s + 5.92·25-s − 14.1·26-s − 1.49·27-s − 8.30·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.145·3-s + 2.42·4-s + 1.47·5-s − 0.269·6-s − 0.646·7-s − 2.64·8-s − 0.978·9-s − 2.73·10-s + 1.52·11-s + 0.353·12-s + 1.49·13-s + 1.19·14-s + 0.215·15-s + 2.46·16-s − 0.190·17-s + 1.81·18-s + 0.269·19-s + 3.58·20-s − 0.0940·21-s − 2.82·22-s − 0.374·23-s − 0.384·24-s + 1.18·25-s − 2.77·26-s − 0.288·27-s − 1.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8780572837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8780572837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.252T + 3T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 5.40T + 13T^{2} \) |
| 17 | \( 1 + 0.784T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 - 7.10T + 29T^{2} \) |
| 31 | \( 1 + 9.66T + 31T^{2} \) |
| 37 | \( 1 + 3.34T + 37T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 - 1.57T + 43T^{2} \) |
| 47 | \( 1 - 4.16T + 47T^{2} \) |
| 53 | \( 1 + 1.42T + 53T^{2} \) |
| 59 | \( 1 - 9.64T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 0.285T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 - 7.56T + 79T^{2} \) |
| 83 | \( 1 + 2.48T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 0.0677T + 97T^{2} \) |
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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
−10.33252087163809259982240458122, −9.511147867875494730166005147942, −8.966901905977217627194341335098, −8.532378731569592699421309085384, −7.11122608790616271868252040268, −6.19945568928341931744043206651, −5.91320127558740713138822347535, −3.46746296393431818013397926608, −2.22007345682876857145305559323, −1.14079133280932028941658238897,
1.14079133280932028941658238897, 2.22007345682876857145305559323, 3.46746296393431818013397926608, 5.91320127558740713138822347535, 6.19945568928341931744043206651, 7.11122608790616271868252040268, 8.532378731569592699421309085384, 8.966901905977217627194341335098, 9.511147867875494730166005147942, 10.33252087163809259982240458122
