| L(s) = 1 | + 0.618·2-s − 3-s − 1.61·4-s − 0.618·6-s − 0.763·7-s − 2.23·8-s + 9-s + 1.61·12-s − 1.23·13-s − 0.472·14-s + 1.85·16-s + 2·17-s + 0.618·18-s + 5·19-s + 0.763·21-s + 2.38·23-s + 2.23·24-s − 0.763·26-s − 27-s + 1.23·28-s − 5.85·29-s + 7·31-s + 5.61·32-s + 1.23·34-s − 1.61·36-s − 7.47·37-s + 3.09·38-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.577·3-s − 0.809·4-s − 0.252·6-s − 0.288·7-s − 0.790·8-s + 0.333·9-s + 0.467·12-s − 0.342·13-s − 0.126·14-s + 0.463·16-s + 0.485·17-s + 0.145·18-s + 1.14·19-s + 0.166·21-s + 0.496·23-s + 0.456·24-s − 0.149·26-s − 0.192·27-s + 0.233·28-s − 1.08·29-s + 1.25·31-s + 0.993·32-s + 0.211·34-s − 0.269·36-s − 1.22·37-s + 0.501·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.198174729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198174729\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 + 0.763T + 7T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 - 2.52T + 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 - 5.09T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 9.38T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 8.09T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
| 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
−7.57368825495617663990819461557, −7.07300950601451656825428542510, −6.05911989980354386544903327572, −5.68172436366251250175226911986, −4.89561538569563748015592263659, −4.45558256406378144687825595793, −3.43102708359153233087557833190, −2.99209793792457470883050491763, −1.57067284450317578710555426985, −0.52997886581542145951744092945,
0.52997886581542145951744092945, 1.57067284450317578710555426985, 2.99209793792457470883050491763, 3.43102708359153233087557833190, 4.45558256406378144687825595793, 4.89561538569563748015592263659, 5.68172436366251250175226911986, 6.05911989980354386544903327572, 7.07300950601451656825428542510, 7.57368825495617663990819461557
