| L(s) = 1 | + 2-s − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s + 11-s − 3·12-s + 14-s − 3·15-s + 16-s + 4·17-s + 6·18-s − 8·19-s + 20-s − 3·21-s + 22-s − 4·23-s − 3·24-s + 25-s − 9·27-s + 28-s + 2·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 0.301·11-s − 0.866·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 1.41·18-s − 1.83·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 1.73·27-s + 0.188·28-s + 0.371·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.377708914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.377708914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 419 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.73318961060870, −14.09730702770670, −13.38532466235516, −13.05191073965689, −12.34793175188604, −12.10566515502967, −11.65700423521963, −10.99315071307599, −10.67640813463889, −10.11165114400790, −9.745635435876501, −8.812023311509465, −8.142708711131244, −7.501234179607942, −6.851512523660849, −6.301783376537587, −5.907722150528568, −5.595895881447233, −4.745657784546937, −4.397786329691160, −3.909093975155204, −2.815303193207094, −2.054698117673528, −1.319622509672178, −0.5789330595222311,
0.5789330595222311, 1.319622509672178, 2.054698117673528, 2.815303193207094, 3.909093975155204, 4.397786329691160, 4.745657784546937, 5.595895881447233, 5.907722150528568, 6.301783376537587, 6.851512523660849, 7.501234179607942, 8.142708711131244, 8.812023311509465, 9.745635435876501, 10.11165114400790, 10.67640813463889, 10.99315071307599, 11.65700423521963, 12.10566515502967, 12.34793175188604, 13.05191073965689, 13.38532466235516, 14.09730702770670, 14.73318961060870
