L(s) = 1 | + 2.45·2-s − 2.32·3-s + 4.00·4-s − 4.02·5-s − 5.70·6-s − 4.56·7-s + 4.92·8-s + 2.42·9-s − 9.85·10-s + 2.49·11-s − 9.33·12-s + 13-s − 11.1·14-s + 9.36·15-s + 4.05·16-s − 6.88·17-s + 5.93·18-s + 1.91·19-s − 16.1·20-s + 10.6·21-s + 6.10·22-s − 4.25·23-s − 11.4·24-s + 11.1·25-s + 2.45·26-s + 1.34·27-s − 18.2·28-s + ⋯ |
L(s) = 1 | + 1.73·2-s − 1.34·3-s + 2.00·4-s − 1.79·5-s − 2.33·6-s − 1.72·7-s + 1.74·8-s + 0.807·9-s − 3.11·10-s + 0.751·11-s − 2.69·12-s + 0.277·13-s − 2.98·14-s + 2.41·15-s + 1.01·16-s − 1.66·17-s + 1.39·18-s + 0.438·19-s − 3.60·20-s + 2.31·21-s + 1.30·22-s − 0.886·23-s − 2.34·24-s + 2.23·25-s + 0.480·26-s + 0.258·27-s − 3.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 3 | \( 1 + 2.32T + 3T^{2} \) |
| 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 11 | \( 1 - 2.49T + 11T^{2} \) |
| 17 | \( 1 + 6.88T + 17T^{2} \) |
| 19 | \( 1 - 1.91T + 19T^{2} \) |
| 23 | \( 1 + 4.25T + 23T^{2} \) |
| 29 | \( 1 - 0.363T + 29T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 - 0.693T + 41T^{2} \) |
| 43 | \( 1 + 0.717T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 0.141T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 0.0239T + 71T^{2} \) |
| 73 | \( 1 - 0.444T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 + 6.57T + 83T^{2} \) |
| 89 | \( 1 - 0.210T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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
−11.29027981059618792314168958507, −10.57949457786809177887114086846, −8.997057142087984197010931738542, −7.33368024270714830046902767342, −6.53599810713980204441958053000, −6.09418072418390488939296432740, −4.69102809175415050573672357216, −3.99436373333675641953020522118, −3.13445990653825280472999601846, 0,
3.13445990653825280472999601846, 3.99436373333675641953020522118, 4.69102809175415050573672357216, 6.09418072418390488939296432740, 6.53599810713980204441958053000, 7.33368024270714830046902767342, 8.997057142087984197010931738542, 10.57949457786809177887114086846, 11.29027981059618792314168958507