L(s) = 1 | + 2-s − 1.56·3-s + 4-s − 5-s − 1.56·6-s − 0.438·7-s + 8-s − 0.561·9-s − 10-s + 11-s − 1.56·12-s − 2·13-s − 0.438·14-s + 1.56·15-s + 16-s − 4.12·17-s − 0.561·18-s + 19-s − 20-s + 0.684·21-s + 22-s + 5.68·23-s − 1.56·24-s + 25-s − 2·26-s + 5.56·27-s − 0.438·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.901·3-s + 0.5·4-s − 0.447·5-s − 0.637·6-s − 0.165·7-s + 0.353·8-s − 0.187·9-s − 0.316·10-s + 0.301·11-s − 0.450·12-s − 0.554·13-s − 0.117·14-s + 0.403·15-s + 0.250·16-s − 0.999·17-s − 0.132·18-s + 0.229·19-s − 0.223·20-s + 0.149·21-s + 0.213·22-s + 1.18·23-s − 0.318·24-s + 0.200·25-s − 0.392·26-s + 1.07·27-s − 0.0828·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 + 0.438T + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 31 | \( 1 - 6.12T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 - 7.68T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 9.12T + 59T^{2} \) |
| 61 | \( 1 + 9.56T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
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
−7.40688436583773574748703499008, −6.48609200975402783303901709164, −6.18576355541795905515003569702, −5.31674636481810275121662786152, −4.69979994823695534585258083865, −4.14960916083854517767813607157, −3.11607214735780044486224253512, −2.47154551228657390587241844980, −1.15285141148719897501403957951, 0,
1.15285141148719897501403957951, 2.47154551228657390587241844980, 3.11607214735780044486224253512, 4.14960916083854517767813607157, 4.69979994823695534585258083865, 5.31674636481810275121662786152, 6.18576355541795905515003569702, 6.48609200975402783303901709164, 7.40688436583773574748703499008