| L(s) = 1 | − 1.98·2-s − 1.72·3-s + 1.93·4-s − 1.05·5-s + 3.41·6-s + 3.75·7-s + 0.132·8-s − 0.0271·9-s + 2.09·10-s + 6.08·11-s − 3.33·12-s − 6.72·13-s − 7.44·14-s + 1.82·15-s − 4.12·16-s − 5.23·17-s + 0.0539·18-s − 0.354·19-s − 2.04·20-s − 6.47·21-s − 12.0·22-s + 6.00·23-s − 0.228·24-s − 3.88·25-s + 13.3·26-s + 5.21·27-s + 7.25·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s − 0.995·3-s + 0.966·4-s − 0.473·5-s + 1.39·6-s + 1.41·7-s + 0.0468·8-s − 0.00906·9-s + 0.663·10-s + 1.83·11-s − 0.962·12-s − 1.86·13-s − 1.99·14-s + 0.470·15-s − 1.03·16-s − 1.26·17-s + 0.0127·18-s − 0.0814·19-s − 0.457·20-s − 1.41·21-s − 2.57·22-s + 1.25·23-s − 0.0465·24-s − 0.776·25-s + 2.61·26-s + 1.00·27-s + 1.37·28-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
| \(L(1)\) |
\(\approx\) |
\(0.4760535321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4760535321\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.98T + 2T^{2} \) |
| 3 | \( 1 + 1.72T + 3T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 7 | \( 1 - 3.75T + 7T^{2} \) |
| 11 | \( 1 - 6.08T + 11T^{2} \) |
| 13 | \( 1 + 6.72T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 0.354T + 19T^{2} \) |
| 23 | \( 1 - 6.00T + 23T^{2} \) |
| 29 | \( 1 - 6.90T + 29T^{2} \) |
| 31 | \( 1 + 5.33T + 31T^{2} \) |
| 37 | \( 1 - 5.05T + 37T^{2} \) |
| 41 | \( 1 - 9.63T + 41T^{2} \) |
| 43 | \( 1 + 0.0907T + 43T^{2} \) |
| 47 | \( 1 + 4.97T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 - 6.12T + 59T^{2} \) |
| 61 | \( 1 - 1.72T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 0.184T + 83T^{2} \) |
| 89 | \( 1 + 7.11T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−10.76788320266700613909558874017, −9.607327195846374853034741716783, −8.954412755970775217488944490111, −8.078547700229649108668016673661, −7.20540311788047990695968038180, −6.49324912056451911790463182677, −4.96289152021834544722611074925, −4.39179835950707295807812093707, −2.13512872343296515296372422759, −0.78666677260535171729752320000,
0.78666677260535171729752320000, 2.13512872343296515296372422759, 4.39179835950707295807812093707, 4.96289152021834544722611074925, 6.49324912056451911790463182677, 7.20540311788047990695968038180, 8.078547700229649108668016673661, 8.954412755970775217488944490111, 9.607327195846374853034741716783, 10.76788320266700613909558874017
