| L(s) = 1 | − 3-s + 3·5-s + 3·7-s − 2·9-s − 13-s − 3·15-s − 7·17-s − 4·19-s − 3·21-s + 4·23-s + 4·25-s + 5·27-s + 4·29-s + 8·31-s + 9·35-s + 7·37-s + 39-s − 2·41-s + 43-s − 6·45-s + 7·47-s + 2·49-s + 7·51-s + 4·53-s + 4·57-s + 14·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1.13·7-s − 2/3·9-s − 0.277·13-s − 0.774·15-s − 1.69·17-s − 0.917·19-s − 0.654·21-s + 0.834·23-s + 4/5·25-s + 0.962·27-s + 0.742·29-s + 1.43·31-s + 1.52·35-s + 1.15·37-s + 0.160·39-s − 0.312·41-s + 0.152·43-s − 0.894·45-s + 1.02·47-s + 2/7·49-s + 0.980·51-s + 0.549·53-s + 0.529·57-s + 1.82·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.034318665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034318665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.688346678043004362720817854167, −8.062957801056656785691485232320, −6.80908640293614540118230030906, −6.38162664221238001798721743035, −5.57492782939857476603569048132, −4.90955045474991307923447518838, −4.30323548470378017580550694537, −2.60212188297765768751674728749, −2.16440348263007722426022041215, −0.886249656884875342153582519243,
0.886249656884875342153582519243, 2.16440348263007722426022041215, 2.60212188297765768751674728749, 4.30323548470378017580550694537, 4.90955045474991307923447518838, 5.57492782939857476603569048132, 6.38162664221238001798721743035, 6.80908640293614540118230030906, 8.062957801056656785691485232320, 8.688346678043004362720817854167
