| L(s) = 1 | − 1.35·2-s + 0.834·4-s + 0.490·5-s + 0.223·8-s + 9-s − 0.665·10-s − 0.803·11-s + 1.89·13-s − 1.13·16-s − 1.97·17-s − 1.35·18-s + 0.409·20-s + 1.08·22-s + 1.09·23-s − 0.758·25-s − 2.56·26-s + 1.57·29-s + 1.31·32-s + 2.67·34-s + 0.834·36-s + 0.109·40-s − 0.670·44-s + 0.490·45-s − 1.48·46-s − 1.75·47-s + 49-s + 1.02·50-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.834·4-s + 0.490·5-s + 0.223·8-s + 9-s − 0.665·10-s − 0.803·11-s + 1.89·13-s − 1.13·16-s − 1.97·17-s − 1.35·18-s + 0.409·20-s + 1.08·22-s + 1.09·23-s − 0.758·25-s − 2.56·26-s + 1.57·29-s + 1.31·32-s + 2.67·34-s + 0.834·36-s + 0.109·40-s − 0.670·44-s + 0.490·45-s − 1.48·46-s − 1.75·47-s + 49-s + 1.02·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5948994314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5948994314\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 919 | \( 1 - T \) |
| good | 2 | \( 1 + 1.35T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.490T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.803T + T^{2} \) |
| 13 | \( 1 - 1.89T + T^{2} \) |
| 17 | \( 1 + 1.97T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.09T + T^{2} \) |
| 29 | \( 1 - 1.57T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.75T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 0.165T + T^{2} \) |
| 61 | \( 1 - 1.09T + T^{2} \) |
| 67 | \( 1 - 1.57T + T^{2} \) |
| 71 | \( 1 - 1.89T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.75T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.17430032326007880501503596787, −9.429227104682446452357465510120, −8.608633931888635595303942601210, −8.132754532760842319288823970684, −6.90808897246507531823778425840, −6.44333654260991692891869875906, −5.01844358155635909368892765662, −4.00298859358926276534466114673, −2.35581872354509480831851415400, −1.23667059665668900838914467309,
1.23667059665668900838914467309, 2.35581872354509480831851415400, 4.00298859358926276534466114673, 5.01844358155635909368892765662, 6.44333654260991692891869875906, 6.90808897246507531823778425840, 8.132754532760842319288823970684, 8.608633931888635595303942601210, 9.429227104682446452357465510120, 10.17430032326007880501503596787
