| L(s) = 1 | + 0.258·2-s + 2.67·3-s − 1.93·4-s − 2.46·5-s + 0.692·6-s + 1.88·7-s − 1.01·8-s + 4.15·9-s − 0.637·10-s + 2.17·11-s − 5.17·12-s + 5.79·13-s + 0.486·14-s − 6.58·15-s + 3.60·16-s + 5.77·17-s + 1.07·18-s − 2.25·19-s + 4.75·20-s + 5.03·21-s + 0.562·22-s − 2.33·23-s − 2.72·24-s + 1.05·25-s + 1.49·26-s + 3.10·27-s − 3.63·28-s + ⋯ |
| L(s) = 1 | + 0.183·2-s + 1.54·3-s − 0.966·4-s − 1.10·5-s + 0.282·6-s + 0.710·7-s − 0.359·8-s + 1.38·9-s − 0.201·10-s + 0.655·11-s − 1.49·12-s + 1.60·13-s + 0.130·14-s − 1.70·15-s + 0.900·16-s + 1.40·17-s + 0.253·18-s − 0.517·19-s + 1.06·20-s + 1.09·21-s + 0.120·22-s − 0.487·23-s − 0.556·24-s + 0.211·25-s + 0.293·26-s + 0.597·27-s − 0.687·28-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
| \(L(1)\) |
\(\approx\) |
\(2.129480415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.129480415\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.258T + 2T^{2} \) |
| 3 | \( 1 - 2.67T + 3T^{2} \) |
| 5 | \( 1 + 2.46T + 5T^{2} \) |
| 7 | \( 1 - 1.88T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 + 2.25T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 9.67T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 + 0.0669T + 53T^{2} \) |
| 59 | \( 1 + 8.20T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 0.238T + 67T^{2} \) |
| 71 | \( 1 - 2.25T + 71T^{2} \) |
| 73 | \( 1 + 9.40T + 73T^{2} \) |
| 79 | \( 1 - 1.71T + 79T^{2} \) |
| 83 | \( 1 + 1.43T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 9.35T + 97T^{2} \) |
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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
−10.48859213906703268210506230728, −9.430518132756196774581114710816, −8.638753132551186059172745897804, −8.181480783618787084504496208097, −7.61317455521247603539238586742, −6.07511155503227349966165091712, −4.61180467888344143505041662948, −3.80480367954781856971379722944, −3.28568403240420321156010004962, −1.36830780614071780582187215135,
1.36830780614071780582187215135, 3.28568403240420321156010004962, 3.80480367954781856971379722944, 4.61180467888344143505041662948, 6.07511155503227349966165091712, 7.61317455521247603539238586742, 8.181480783618787084504496208097, 8.638753132551186059172745897804, 9.430518132756196774581114710816, 10.48859213906703268210506230728
