| L(s) = 1 | + 2.81·2-s + 1.69·3-s + 5.94·4-s − 2.73·5-s + 4.77·6-s − 1.24·7-s + 11.1·8-s − 0.129·9-s − 7.69·10-s − 0.667·11-s + 10.0·12-s + 0.636·13-s − 3.51·14-s − 4.62·15-s + 19.4·16-s − 1.78·17-s − 0.363·18-s − 5.81·19-s − 16.2·20-s − 2.11·21-s − 1.88·22-s − 1.72·23-s + 18.8·24-s + 2.45·25-s + 1.79·26-s − 5.30·27-s − 7.40·28-s + ⋯ |
| L(s) = 1 | + 1.99·2-s + 0.978·3-s + 2.97·4-s − 1.22·5-s + 1.94·6-s − 0.471·7-s + 3.92·8-s − 0.0430·9-s − 2.43·10-s − 0.201·11-s + 2.90·12-s + 0.176·13-s − 0.939·14-s − 1.19·15-s + 4.85·16-s − 0.433·17-s − 0.0856·18-s − 1.33·19-s − 3.62·20-s − 0.461·21-s − 0.400·22-s − 0.359·23-s + 3.84·24-s + 0.491·25-s + 0.351·26-s − 1.02·27-s − 1.40·28-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
| \(L(1)\) |
\(\approx\) |
\(5.161329785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.161329785\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.81T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 + 0.667T + 11T^{2} \) |
| 13 | \( 1 - 0.636T + 13T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 + 5.81T + 19T^{2} \) |
| 23 | \( 1 + 1.72T + 23T^{2} \) |
| 29 | \( 1 - 5.11T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 0.542T + 43T^{2} \) |
| 47 | \( 1 + 7.59T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 5.40T + 59T^{2} \) |
| 61 | \( 1 - 7.79T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 2.62T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 0.970T + 89T^{2} \) |
| 97 | \( 1 + 6.09T + 97T^{2} \) |
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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
−11.15459682297677273126072443450, −10.04817556198669582413392793804, −8.422700896106344999246726650314, −7.84317840437518248209202116957, −6.82594173650390036063897957689, −6.03634443375816023834731850774, −4.67599414915801600675596665872, −3.94794609656790578519764239538, −3.17931953570429837273854830600, −2.27544098013980643947967490297,
2.27544098013980643947967490297, 3.17931953570429837273854830600, 3.94794609656790578519764239538, 4.67599414915801600675596665872, 6.03634443375816023834731850774, 6.82594173650390036063897957689, 7.84317840437518248209202116957, 8.422700896106344999246726650314, 10.04817556198669582413392793804, 11.15459682297677273126072443450
