L(s) = 1 | + 3-s + 1.77·5-s + 7-s + 9-s + 6.49·11-s − 13-s + 1.77·15-s − 2.94·17-s + 4.83·19-s + 21-s + 5.77·23-s − 1.83·25-s + 27-s − 2.83·29-s − 6.27·31-s + 6.49·33-s + 1.77·35-s + 9.55·37-s − 39-s − 3.05·41-s − 2.71·43-s + 1.77·45-s + 8.71·47-s + 49-s − 2.94·51-s + 6.39·53-s + 11.5·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.795·5-s + 0.377·7-s + 0.333·9-s + 1.95·11-s − 0.277·13-s + 0.459·15-s − 0.713·17-s + 1.10·19-s + 0.218·21-s + 1.20·23-s − 0.367·25-s + 0.192·27-s − 0.526·29-s − 1.12·31-s + 1.13·33-s + 0.300·35-s + 1.57·37-s − 0.160·39-s − 0.477·41-s − 0.414·43-s + 0.265·45-s + 1.27·47-s + 0.142·49-s − 0.411·51-s + 0.878·53-s + 1.55·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.511969577\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.511969577\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 1.77T + 5T^{2} \) |
| 11 | \( 1 - 6.49T + 11T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 + 2.83T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 + 2.71T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 - 6.39T + 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 61 | \( 1 - 3.88T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 + 7.66T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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
−8.651452382011568202377776879736, −7.44141575429811491398048602916, −7.08221311096602028718017383191, −6.16855091874199794146624159282, −5.49716899698305666801874183200, −4.49574862830474558517597382260, −3.82640383569834414601557828135, −2.86524636635124487476629512724, −1.86481882783532466984811163157, −1.14921377970202771654698617056,
1.14921377970202771654698617056, 1.86481882783532466984811163157, 2.86524636635124487476629512724, 3.82640383569834414601557828135, 4.49574862830474558517597382260, 5.49716899698305666801874183200, 6.16855091874199794146624159282, 7.08221311096602028718017383191, 7.44141575429811491398048602916, 8.651452382011568202377776879736