| L(s) = 1 | + 0.735·2-s − 1.45·4-s + 1.54·5-s + 0.781·7-s − 2.54·8-s + 1.13·10-s − 11-s − 1.76·13-s + 0.575·14-s + 1.04·16-s − 7.26·17-s + 6.84·19-s − 2.25·20-s − 0.735·22-s + 23-s − 2.61·25-s − 1.30·26-s − 1.14·28-s − 5.92·29-s − 9.96·31-s + 5.85·32-s − 5.34·34-s + 1.20·35-s + 0.378·37-s + 5.03·38-s − 3.92·40-s − 1.53·41-s + ⋯ |
| L(s) = 1 | + 0.520·2-s − 0.729·4-s + 0.690·5-s + 0.295·7-s − 0.899·8-s + 0.359·10-s − 0.301·11-s − 0.490·13-s + 0.153·14-s + 0.261·16-s − 1.76·17-s + 1.56·19-s − 0.503·20-s − 0.156·22-s + 0.208·23-s − 0.523·25-s − 0.255·26-s − 0.215·28-s − 1.10·29-s − 1.79·31-s + 1.03·32-s − 0.916·34-s + 0.204·35-s + 0.0622·37-s + 0.816·38-s − 0.621·40-s − 0.239·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2277 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2277 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.735T + 2T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 - 0.781T + 7T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 7.26T + 17T^{2} \) |
| 19 | \( 1 - 6.84T + 19T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 + 9.96T + 31T^{2} \) |
| 37 | \( 1 - 0.378T + 37T^{2} \) |
| 41 | \( 1 + 1.53T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 3.89T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 3.69T + 71T^{2} \) |
| 73 | \( 1 + 2.10T + 73T^{2} \) |
| 79 | \( 1 + 2.20T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 4.59T + 97T^{2} \) |
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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
−8.880633780599066546744512289258, −7.80517066528617714042195548412, −7.09215127671009378432024010702, −5.97361095457430291454652307575, −5.41160205324747458072386886101, −4.70639212953795368502367923327, −3.83826149579392056951730324511, −2.79632091215673012273795461562, −1.72193358405821942957389150410, 0,
1.72193358405821942957389150410, 2.79632091215673012273795461562, 3.83826149579392056951730324511, 4.70639212953795368502367923327, 5.41160205324747458072386886101, 5.97361095457430291454652307575, 7.09215127671009378432024010702, 7.80517066528617714042195548412, 8.880633780599066546744512289258
