| L(s) = 1 | − 2.52·2-s + 4.37·4-s + 2.37·5-s + 0.792·7-s − 5.98·8-s − 5.98·10-s − 4.10·13-s − 2·14-s + 6.37·16-s + 5.98·17-s + 4.25·19-s + 10.3·20-s − 2·23-s + 0.627·25-s + 10.3·26-s + 3.46·28-s + 2.52·29-s + 7.37·31-s − 4.10·32-s − 15.1·34-s + 1.87·35-s + 5·37-s − 10.7·38-s − 14.2·40-s + 5.69·41-s − 6.63·43-s + 5.04·46-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 2.18·4-s + 1.06·5-s + 0.299·7-s − 2.11·8-s − 1.89·10-s − 1.13·13-s − 0.534·14-s + 1.59·16-s + 1.45·17-s + 0.976·19-s + 2.31·20-s − 0.417·23-s + 0.125·25-s + 2.03·26-s + 0.654·28-s + 0.468·29-s + 1.32·31-s − 0.726·32-s − 2.59·34-s + 0.317·35-s + 0.821·37-s − 1.74·38-s − 2.24·40-s + 0.889·41-s − 1.01·43-s + 0.744·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9024849307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9024849307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 0.792T + 7T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 2.67T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 + 0.744T + 71T^{2} \) |
| 73 | \( 1 - 7.42T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 - 6.37T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−9.893230379815947881317288060478, −9.345397197793378863612865025123, −8.131298011531464336897081889546, −7.76072687608090308112669813050, −6.76118927664504223906738166990, −5.91446376592731323298599632085, −4.93674085223662595690386766880, −3.01886103306354810875935909790, −2.05619132340002841063058710101, −0.984831541696738158487430064108,
0.984831541696738158487430064108, 2.05619132340002841063058710101, 3.01886103306354810875935909790, 4.93674085223662595690386766880, 5.91446376592731323298599632085, 6.76118927664504223906738166990, 7.76072687608090308112669813050, 8.131298011531464336897081889546, 9.345397197793378863612865025123, 9.893230379815947881317288060478
