| L(s) = 1 | − 1.53·2-s − 0.943·3-s + 0.356·4-s + 1.44·6-s + 7-s + 2.52·8-s − 2.11·9-s + 3.95·11-s − 0.336·12-s − 5.22·13-s − 1.53·14-s − 4.58·16-s − 3.97·17-s + 3.23·18-s − 4.56·19-s − 0.943·21-s − 6.06·22-s + 23-s − 2.37·24-s + 8.01·26-s + 4.81·27-s + 0.356·28-s + 5.14·29-s − 0.386·31-s + 1.99·32-s − 3.72·33-s + 6.10·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.544·3-s + 0.178·4-s + 0.591·6-s + 0.377·7-s + 0.892·8-s − 0.703·9-s + 1.19·11-s − 0.0970·12-s − 1.44·13-s − 0.410·14-s − 1.14·16-s − 0.964·17-s + 0.763·18-s − 1.04·19-s − 0.205·21-s − 1.29·22-s + 0.208·23-s − 0.485·24-s + 1.57·26-s + 0.927·27-s + 0.0673·28-s + 0.954·29-s − 0.0693·31-s + 0.352·32-s − 0.648·33-s + 1.04·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4613312905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4613312905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 + 0.943T + 3T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 + 5.22T + 13T^{2} \) |
| 17 | \( 1 + 3.97T + 17T^{2} \) |
| 19 | \( 1 + 4.56T + 19T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 + 0.386T + 31T^{2} \) |
| 37 | \( 1 + 9.97T + 37T^{2} \) |
| 41 | \( 1 - 7.11T + 41T^{2} \) |
| 43 | \( 1 + 5.94T + 43T^{2} \) |
| 47 | \( 1 + 2.97T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 1.77T + 59T^{2} \) |
| 61 | \( 1 - 1.66T + 61T^{2} \) |
| 67 | \( 1 + 4.98T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 9.53T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.623391527556763649747711440848, −7.913228305056972597874440071187, −6.87829394469931768805553926960, −6.63316570862384228711960136725, −5.42282255122919597218244334330, −4.71511196135411681536057810841, −4.05397478858363182435965593249, −2.61229986675517109827059447578, −1.70672890380919246776819176410, −0.47178213308744556388659672139,
0.47178213308744556388659672139, 1.70672890380919246776819176410, 2.61229986675517109827059447578, 4.05397478858363182435965593249, 4.71511196135411681536057810841, 5.42282255122919597218244334330, 6.63316570862384228711960136725, 6.87829394469931768805553926960, 7.913228305056972597874440071187, 8.623391527556763649747711440848
