| L(s) = 1 | − 1.73·2-s + 0.999·4-s − 5-s + 3.46·7-s + 1.73·8-s + 1.73·10-s − 5.99·14-s − 5·16-s + 6.92·17-s − 6.92·19-s − 0.999·20-s − 6·23-s + 25-s + 3.46·28-s + 4·31-s + 5.19·32-s − 11.9·34-s − 3.46·35-s + 10·37-s + 11.9·38-s − 1.73·40-s − 6.92·41-s − 3.46·43-s + 10.3·46-s + 6·47-s + 4.99·49-s − 1.73·50-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s − 0.447·5-s + 1.30·7-s + 0.612·8-s + 0.547·10-s − 1.60·14-s − 1.25·16-s + 1.68·17-s − 1.58·19-s − 0.223·20-s − 1.25·23-s + 0.200·25-s + 0.654·28-s + 0.718·31-s + 0.918·32-s − 2.05·34-s − 0.585·35-s + 1.64·37-s + 1.94·38-s − 0.273·40-s − 1.08·41-s − 0.528·43-s + 1.53·46-s + 0.875·47-s + 0.714·49-s − 0.244·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9924420056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9924420056\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6.92T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.175415925003182435082487654810, −7.86111165863241733429610027212, −7.08465732010925758742449504209, −6.15404025776713912084916599290, −5.22868219401025558361280725709, −4.44607171187759283187245243385, −3.83420951967696484193539125086, −2.43447866653361366300142562393, −1.60625456357467009633006163365, −0.68278769191077957690050065765,
0.68278769191077957690050065765, 1.60625456357467009633006163365, 2.43447866653361366300142562393, 3.83420951967696484193539125086, 4.44607171187759283187245243385, 5.22868219401025558361280725709, 6.15404025776713912084916599290, 7.08465732010925758742449504209, 7.86111165863241733429610027212, 8.175415925003182435082487654810
