| L(s) = 1 | + 2.45·2-s − 3-s + 4.02·4-s − 2.45·6-s + 2.54·7-s + 4.96·8-s + 9-s + 1.37·11-s − 4.02·12-s − 0.542·13-s + 6.24·14-s + 4.14·16-s + 0.0592·17-s + 2.45·18-s − 0.567·19-s − 2.54·21-s + 3.36·22-s + 8.45·23-s − 4.96·24-s − 1.33·26-s − 27-s + 10.2·28-s − 29-s + 7.33·31-s + 0.238·32-s − 1.37·33-s + 0.145·34-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.577·3-s + 2.01·4-s − 1.00·6-s + 0.960·7-s + 1.75·8-s + 0.333·9-s + 0.413·11-s − 1.16·12-s − 0.150·13-s + 1.66·14-s + 1.03·16-s + 0.0143·17-s + 0.578·18-s − 0.130·19-s − 0.554·21-s + 0.717·22-s + 1.76·23-s − 1.01·24-s − 0.261·26-s − 0.192·27-s + 1.93·28-s − 0.185·29-s + 1.31·31-s + 0.0421·32-s − 0.238·33-s + 0.0249·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.917830723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.917830723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 7 | \( 1 - 2.54T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 0.542T + 13T^{2} \) |
| 17 | \( 1 - 0.0592T + 17T^{2} \) |
| 19 | \( 1 + 0.567T + 19T^{2} \) |
| 23 | \( 1 - 8.45T + 23T^{2} \) |
| 31 | \( 1 - 7.33T + 31T^{2} \) |
| 37 | \( 1 + 5.86T + 37T^{2} \) |
| 41 | \( 1 + 4.87T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 7.49T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 - 0.0372T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 2.45T + 71T^{2} \) |
| 73 | \( 1 - 8.17T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 3.11T + 97T^{2} \) |
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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
−9.002461016740966087350085302359, −8.077253353773038813669726944186, −6.98270949084883562250686854591, −6.66023908739923206821911248237, −5.54540853659978433833497111149, −5.06528465776223024348963560731, −4.41711193161120928868193673383, −3.52596989819863437610363183622, −2.46731251800110340400317577268, −1.31692048472570001891339844102,
1.31692048472570001891339844102, 2.46731251800110340400317577268, 3.52596989819863437610363183622, 4.41711193161120928868193673383, 5.06528465776223024348963560731, 5.54540853659978433833497111149, 6.66023908739923206821911248237, 6.98270949084883562250686854591, 8.077253353773038813669726944186, 9.002461016740966087350085302359
