L(s) = 1 | + 2-s − 2.44·3-s + 4-s − 5-s − 2.44·6-s + 7-s + 8-s + 2.99·9-s − 10-s + 4.44·11-s − 2.44·12-s − 13-s + 14-s + 2.44·15-s + 16-s + 2.44·17-s + 2.99·18-s + 5.89·19-s − 20-s − 2.44·21-s + 4.44·22-s − 0.449·23-s − 2.44·24-s + 25-s − 26-s + 28-s + 9.44·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s − 0.447·5-s − 0.999·6-s + 0.377·7-s + 0.353·8-s + 0.999·9-s − 0.316·10-s + 1.34·11-s − 0.707·12-s − 0.277·13-s + 0.267·14-s + 0.632·15-s + 0.250·16-s + 0.594·17-s + 0.707·18-s + 1.35·19-s − 0.223·20-s − 0.534·21-s + 0.948·22-s − 0.0937·23-s − 0.499·24-s + 0.200·25-s − 0.196·26-s + 0.188·28-s + 1.75·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.409916214\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409916214\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 5.89T + 19T^{2} \) |
| 23 | \( 1 + 0.449T + 23T^{2} \) |
| 29 | \( 1 - 9.44T + 29T^{2} \) |
| 31 | \( 1 + 0.550T + 31T^{2} \) |
| 37 | \( 1 - 0.449T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 47 | \( 1 - 2.44T + 47T^{2} \) |
| 53 | \( 1 + 5.79T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 + 5.44T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 7.44T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 6.24T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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
−11.57886774018344567188528681772, −10.59063867334133507634046224733, −9.638648153953077654907421399715, −8.221101802109247274942236298765, −7.04184770716146158232899477000, −6.35761167740484506599188413863, −5.30009835906302679686463105854, −4.58869417002471714506163121979, −3.35615617589695242924407440320, −1.20518685442786568611388671417,
1.20518685442786568611388671417, 3.35615617589695242924407440320, 4.58869417002471714506163121979, 5.30009835906302679686463105854, 6.35761167740484506599188413863, 7.04184770716146158232899477000, 8.221101802109247274942236298765, 9.638648153953077654907421399715, 10.59063867334133507634046224733, 11.57886774018344567188528681772