| L(s) = 1 | − 2-s + 0.512·3-s + 4-s − 1.04·5-s − 0.512·6-s − 2.90·7-s − 8-s − 2.73·9-s + 1.04·10-s + 2.34·11-s + 0.512·12-s + 4.24·13-s + 2.90·14-s − 0.537·15-s + 16-s + 7.66·17-s + 2.73·18-s + 5.81·19-s − 1.04·20-s − 1.48·21-s − 2.34·22-s + 6.53·23-s − 0.512·24-s − 3.89·25-s − 4.24·26-s − 2.93·27-s − 2.90·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.295·3-s + 0.5·4-s − 0.469·5-s − 0.209·6-s − 1.09·7-s − 0.353·8-s − 0.912·9-s + 0.331·10-s + 0.706·11-s + 0.147·12-s + 1.17·13-s + 0.776·14-s − 0.138·15-s + 0.250·16-s + 1.85·17-s + 0.645·18-s + 1.33·19-s − 0.234·20-s − 0.324·21-s − 0.499·22-s + 1.36·23-s − 0.104·24-s − 0.779·25-s − 0.833·26-s − 0.565·27-s − 0.549·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9976014903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9976014903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 - 0.512T + 3T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 7.66T + 17T^{2} \) |
| 19 | \( 1 - 5.81T + 19T^{2} \) |
| 23 | \( 1 - 6.53T + 23T^{2} \) |
| 29 | \( 1 - 0.119T + 29T^{2} \) |
| 31 | \( 1 - 3.75T + 31T^{2} \) |
| 37 | \( 1 + 3.27T + 37T^{2} \) |
| 41 | \( 1 - 4.46T + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 0.549T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 - 9.15T + 73T^{2} \) |
| 79 | \( 1 - 0.426T + 79T^{2} \) |
| 83 | \( 1 - 7.84T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
| 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
−10.74963236169591012154453595844, −9.694706454794153445319860383700, −9.148427629012751799882000886355, −8.224718373336695943917140103840, −7.41173367901010032285109530688, −6.34434855145336749904464772859, −5.51794890639728733621034819689, −3.56124269324000870517423657380, −3.09831440209376624748036504763, −1.02463146886105758623892274804,
1.02463146886105758623892274804, 3.09831440209376624748036504763, 3.56124269324000870517423657380, 5.51794890639728733621034819689, 6.34434855145336749904464772859, 7.41173367901010032285109530688, 8.224718373336695943917140103840, 9.148427629012751799882000886355, 9.694706454794153445319860383700, 10.74963236169591012154453595844
