L(s) = 1 | + 0.210·3-s + 0.593·5-s − 1.60·7-s − 2.95·9-s − 2.16·11-s + 2.74·13-s + 0.125·15-s + 17-s + 0.223·19-s − 0.338·21-s + 1.30·23-s − 4.64·25-s − 1.25·27-s + 9.53·29-s + 5.20·31-s − 0.455·33-s − 0.953·35-s − 0.872·37-s + 0.577·39-s − 10.2·41-s + 0.790·43-s − 1.75·45-s + 1.57·47-s − 4.41·49-s + 0.210·51-s − 5.39·53-s − 1.28·55-s + ⋯ |
L(s) = 1 | + 0.121·3-s + 0.265·5-s − 0.607·7-s − 0.985·9-s − 0.651·11-s + 0.759·13-s + 0.0322·15-s + 0.242·17-s + 0.0512·19-s − 0.0739·21-s + 0.271·23-s − 0.929·25-s − 0.241·27-s + 1.77·29-s + 0.934·31-s − 0.0793·33-s − 0.161·35-s − 0.143·37-s + 0.0925·39-s − 1.60·41-s + 0.120·43-s − 0.261·45-s + 0.229·47-s − 0.631·49-s + 0.0295·51-s − 0.740·53-s − 0.172·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.585571023\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.585571023\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 0.210T + 3T^{2} \) |
| 5 | \( 1 - 0.593T + 5T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 19 | \( 1 - 0.223T + 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 - 9.53T + 29T^{2} \) |
| 31 | \( 1 - 5.20T + 31T^{2} \) |
| 37 | \( 1 + 0.872T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 0.790T + 43T^{2} \) |
| 47 | \( 1 - 1.57T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 4.98T + 73T^{2} \) |
| 79 | \( 1 + 8.57T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 6.17T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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
−8.323390824861054961826772409984, −8.008498291199830209962618604307, −6.77370599019962831961461626203, −6.29771630226767183179612377179, −5.51146560947157121600046770341, −4.81319132090473105377292740527, −3.61343456344973183430877533125, −3.00942709169836139025840191745, −2.12078852270870367284418016668, −0.69788465123315019393826929409,
0.69788465123315019393826929409, 2.12078852270870367284418016668, 3.00942709169836139025840191745, 3.61343456344973183430877533125, 4.81319132090473105377292740527, 5.51146560947157121600046770341, 6.29771630226767183179612377179, 6.77370599019962831961461626203, 8.008498291199830209962618604307, 8.323390824861054961826772409984