| L(s) = 1 | + 2-s − 2.54·3-s + 4-s + 5-s − 2.54·6-s + 7-s + 8-s + 3.46·9-s + 10-s − 2.54·12-s − 6.42·13-s + 14-s − 2.54·15-s + 16-s − 3.95·17-s + 3.46·18-s + 2.46·19-s + 20-s − 2.54·21-s + 4.99·23-s − 2.54·24-s + 25-s − 6.42·26-s − 1.19·27-s + 28-s − 9.04·29-s − 2.54·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.46·3-s + 0.5·4-s + 0.447·5-s − 1.03·6-s + 0.377·7-s + 0.353·8-s + 1.15·9-s + 0.316·10-s − 0.734·12-s − 1.78·13-s + 0.267·14-s − 0.656·15-s + 0.250·16-s − 0.958·17-s + 0.817·18-s + 0.566·19-s + 0.223·20-s − 0.555·21-s + 1.04·23-s − 0.519·24-s + 0.200·25-s − 1.25·26-s − 0.229·27-s + 0.188·28-s − 1.67·29-s − 0.464·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.54T + 3T^{2} \) |
| 13 | \( 1 + 6.42T + 13T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 + 9.04T + 29T^{2} \) |
| 31 | \( 1 - 7.88T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 + 8.37T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 + 3.59T + 61T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 + 5.90T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 6.21T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
| 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
−7.16108861985579202703026156878, −6.64486338621118466527963907519, −5.86964229828567738771834607749, −5.36588050435770427081491629219, −4.70044080302775976862849784172, −4.40141239893183279096390549541, −3.01040462017020156443432917075, −2.26913551880451943458853095703, −1.21652336148100754141193278792, 0,
1.21652336148100754141193278792, 2.26913551880451943458853095703, 3.01040462017020156443432917075, 4.40141239893183279096390549541, 4.70044080302775976862849784172, 5.36588050435770427081491629219, 5.86964229828567738771834607749, 6.64486338621118466527963907519, 7.16108861985579202703026156878
