| L(s) = 1 | − 2-s + 4-s − 0.437·5-s + 3.40·7-s − 8-s + 0.437·10-s + 1.43·11-s + 4.53·13-s − 3.40·14-s + 16-s − 4.02·17-s + 1.84·19-s − 0.437·20-s − 1.43·22-s − 3.77·23-s − 4.80·25-s − 4.53·26-s + 3.40·28-s + 9.10·29-s − 3.17·31-s − 32-s + 4.02·34-s − 1.48·35-s + 9.61·37-s − 1.84·38-s + 0.437·40-s + 3.87·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.195·5-s + 1.28·7-s − 0.353·8-s + 0.138·10-s + 0.432·11-s + 1.25·13-s − 0.909·14-s + 0.250·16-s − 0.977·17-s + 0.423·19-s − 0.0977·20-s − 0.305·22-s − 0.788·23-s − 0.961·25-s − 0.889·26-s + 0.643·28-s + 1.69·29-s − 0.569·31-s − 0.176·32-s + 0.690·34-s − 0.251·35-s + 1.58·37-s − 0.299·38-s + 0.0691·40-s + 0.604·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.709954259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709954259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 0.437T + 5T^{2} \) |
| 7 | \( 1 - 3.40T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 + 3.77T + 23T^{2} \) |
| 29 | \( 1 - 9.10T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 - 9.61T + 37T^{2} \) |
| 41 | \( 1 - 3.87T + 41T^{2} \) |
| 43 | \( 1 - 9.48T + 43T^{2} \) |
| 47 | \( 1 - 3.28T + 47T^{2} \) |
| 53 | \( 1 - 8.67T + 53T^{2} \) |
| 59 | \( 1 + 1.42T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 4.98T + 79T^{2} \) |
| 83 | \( 1 - 6.21T + 83T^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
| 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
−8.402108801801078318285355269670, −7.899703409556514682092093429568, −7.19833054543701225168159821925, −6.22040618204368760401112798825, −5.71990044989033857376791913749, −4.46589477215543289380073640905, −4.02358773705492132390653859585, −2.71547289768375995695091910908, −1.75277769287851410695756871004, −0.888318930650294845475192220116,
0.888318930650294845475192220116, 1.75277769287851410695756871004, 2.71547289768375995695091910908, 4.02358773705492132390653859585, 4.46589477215543289380073640905, 5.71990044989033857376791913749, 6.22040618204368760401112798825, 7.19833054543701225168159821925, 7.899703409556514682092093429568, 8.402108801801078318285355269670
