| L(s) = 1 | + 2-s + 5·3-s − 7·4-s + 5·5-s + 5·6-s + 22·7-s − 15·8-s − 2·9-s + 5·10-s + 9·11-s − 35·12-s − 54·13-s + 22·14-s + 25·15-s + 41·16-s − 54·17-s − 2·18-s − 35·20-s + 110·21-s + 9·22-s − 92·23-s − 75·24-s + 25·25-s − 54·26-s − 145·27-s − 154·28-s + 134·29-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.962·3-s − 7/8·4-s + 0.447·5-s + 0.340·6-s + 1.18·7-s − 0.662·8-s − 0.0740·9-s + 0.158·10-s + 0.246·11-s − 0.841·12-s − 1.15·13-s + 0.419·14-s + 0.430·15-s + 0.640·16-s − 0.770·17-s − 0.0261·18-s − 0.391·20-s + 1.14·21-s + 0.0872·22-s − 0.834·23-s − 0.637·24-s + 1/5·25-s − 0.407·26-s − 1.03·27-s − 1.03·28-s + 0.858·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.358267159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.358267159\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 23 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 134 T + p^{3} T^{2} \) |
| 31 | \( 1 - 252 T + p^{3} T^{2} \) |
| 37 | \( 1 - 236 T + p^{3} T^{2} \) |
| 41 | \( 1 - 243 T + p^{3} T^{2} \) |
| 43 | \( 1 - 496 T + p^{3} T^{2} \) |
| 47 | \( 1 - 502 T + p^{3} T^{2} \) |
| 53 | \( 1 + 62 T + p^{3} T^{2} \) |
| 59 | \( 1 + 681 T + p^{3} T^{2} \) |
| 61 | \( 1 + 142 T + p^{3} T^{2} \) |
| 67 | \( 1 + 55 T + p^{3} T^{2} \) |
| 71 | \( 1 - 974 T + p^{3} T^{2} \) |
| 73 | \( 1 - 695 T + p^{3} T^{2} \) |
| 79 | \( 1 - 736 T + p^{3} T^{2} \) |
| 83 | \( 1 + 63 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1167 T + p^{3} T^{2} \) |
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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
−9.040533006961932858533980309898, −8.029923844005332646333265898395, −7.79863349521813762154467092003, −6.39411108851997112184417280119, −5.51525144840996978057623173479, −4.59661043956248809532691572322, −4.15643462861115143990029804427, −2.79581992672832109505528291357, −2.19107390987726155383446580028, −0.78314050003126340393231316815,
0.78314050003126340393231316815, 2.19107390987726155383446580028, 2.79581992672832109505528291357, 4.15643462861115143990029804427, 4.59661043956248809532691572322, 5.51525144840996978057623173479, 6.39411108851997112184417280119, 7.79863349521813762154467092003, 8.029923844005332646333265898395, 9.040533006961932858533980309898
