| L(s) = 1 | + 5-s + 7-s + 6·11-s + 6·13-s + 2·17-s + 7·19-s + 23-s − 4·25-s − 2·29-s + 10·31-s + 35-s − 6·37-s + 8·41-s − 10·43-s − 8·47-s + 49-s − 2·53-s + 6·55-s + 7·61-s + 6·65-s − 12·67-s − 15·71-s − 2·73-s + 6·77-s + 79-s − 12·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.80·11-s + 1.66·13-s + 0.485·17-s + 1.60·19-s + 0.208·23-s − 4/5·25-s − 0.371·29-s + 1.79·31-s + 0.169·35-s − 0.986·37-s + 1.24·41-s − 1.52·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s + 0.809·55-s + 0.896·61-s + 0.744·65-s − 1.46·67-s − 1.78·71-s − 0.234·73-s + 0.683·77-s + 0.112·79-s − 1.31·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.054793034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.054793034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.460095941009109367604154676152, −7.62853361422202650085332123774, −6.75778009886636317888600581021, −6.13984952667230449829780014899, −5.55782610661352515541339492471, −4.53630096095552390250482974797, −3.72462631167540553827048166168, −3.08371928803854652319736627806, −1.54842755853086999230961203081, −1.18913645458013163821193224515,
1.18913645458013163821193224515, 1.54842755853086999230961203081, 3.08371928803854652319736627806, 3.72462631167540553827048166168, 4.53630096095552390250482974797, 5.55782610661352515541339492471, 6.13984952667230449829780014899, 6.75778009886636317888600581021, 7.62853361422202650085332123774, 8.460095941009109367604154676152
