| L(s) = 1 | − 1.45·2-s + 2.53·3-s + 0.116·4-s + 5-s − 3.68·6-s − 0.554·7-s + 2.74·8-s + 3.41·9-s − 1.45·10-s − 0.0423·11-s + 0.294·12-s + 3.60·13-s + 0.806·14-s + 2.53·15-s − 4.21·16-s − 2.41·17-s − 4.96·18-s − 7.73·19-s + 0.116·20-s − 1.40·21-s + 0.0616·22-s + 4.31·23-s + 6.93·24-s + 25-s − 5.24·26-s + 1.04·27-s − 0.0645·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 1.46·3-s + 0.0582·4-s + 0.447·5-s − 1.50·6-s − 0.209·7-s + 0.968·8-s + 1.13·9-s − 0.460·10-s − 0.0127·11-s + 0.0851·12-s + 0.999·13-s + 0.215·14-s + 0.653·15-s − 1.05·16-s − 0.585·17-s − 1.16·18-s − 1.77·19-s + 0.0260·20-s − 0.306·21-s + 0.0131·22-s + 0.898·23-s + 1.41·24-s + 0.200·25-s − 1.02·26-s + 0.200·27-s − 0.0122·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.935123342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.935123342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 - 2.53T + 3T^{2} \) |
| 7 | \( 1 + 0.554T + 7T^{2} \) |
| 11 | \( 1 + 0.0423T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 + 7.73T + 19T^{2} \) |
| 23 | \( 1 - 4.31T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 37 | \( 1 + 3.28T + 37T^{2} \) |
| 41 | \( 1 - 9.41T + 41T^{2} \) |
| 43 | \( 1 + 4.62T + 43T^{2} \) |
| 47 | \( 1 - 3.14T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.450138030698922344716648874716, −8.055188338496276486583225013902, −6.93670668606049987861316838010, −6.55250805910308915307439533854, −5.31784189117462934797775177425, −4.27365089911310745636202711804, −3.70782021113861175691843346814, −2.55248633109996931077839361539, −1.97787600416764416885520487717, −0.853672526227313886249196019613,
0.853672526227313886249196019613, 1.97787600416764416885520487717, 2.55248633109996931077839361539, 3.70782021113861175691843346814, 4.27365089911310745636202711804, 5.31784189117462934797775177425, 6.55250805910308915307439533854, 6.93670668606049987861316838010, 8.055188338496276486583225013902, 8.450138030698922344716648874716
