L(s) = 1 | − 2-s − 1.20·3-s + 4-s + 0.0448·5-s + 1.20·6-s − 2.73·7-s − 8-s − 1.53·9-s − 0.0448·10-s + 2.04·11-s − 1.20·12-s + 1.85·13-s + 2.73·14-s − 0.0542·15-s + 16-s + 1.92·17-s + 1.53·18-s − 19-s + 0.0448·20-s + 3.31·21-s − 2.04·22-s − 2.83·23-s + 1.20·24-s − 4.99·25-s − 1.85·26-s + 5.48·27-s − 2.73·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.698·3-s + 0.5·4-s + 0.0200·5-s + 0.493·6-s − 1.03·7-s − 0.353·8-s − 0.511·9-s − 0.0141·10-s + 0.615·11-s − 0.349·12-s + 0.515·13-s + 0.731·14-s − 0.0140·15-s + 0.250·16-s + 0.466·17-s + 0.362·18-s − 0.229·19-s + 0.0100·20-s + 0.723·21-s − 0.435·22-s − 0.590·23-s + 0.246·24-s − 0.999·25-s − 0.364·26-s + 1.05·27-s − 0.517·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5820626194\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5820626194\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 + 1.20T + 3T^{2} \) |
| 5 | \( 1 - 0.0448T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 2.04T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 23 | \( 1 + 2.83T + 23T^{2} \) |
| 29 | \( 1 - 6.47T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 + 2.03T + 37T^{2} \) |
| 41 | \( 1 - 4.98T + 41T^{2} \) |
| 43 | \( 1 + 0.0680T + 43T^{2} \) |
| 47 | \( 1 - 9.41T + 47T^{2} \) |
| 53 | \( 1 + 9.40T + 53T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 + 7.94T + 61T^{2} \) |
| 67 | \( 1 - 1.66T + 67T^{2} \) |
| 71 | \( 1 + 0.505T + 71T^{2} \) |
| 73 | \( 1 - 0.785T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 1.46T + 89T^{2} \) |
| 97 | \( 1 + 5.36T + 97T^{2} \) |
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
−7.86842127511508525968543300761, −7.07606662962461093646538640068, −6.32376885367540155077894281945, −6.03417713496676676718654486671, −5.32965192499966450627569329263, −4.17204937211163430221836374047, −3.44357804957850299081342389637, −2.63413618315273181332446806801, −1.51878791468406794723249669037, −0.44412750540045851463990947679,
0.44412750540045851463990947679, 1.51878791468406794723249669037, 2.63413618315273181332446806801, 3.44357804957850299081342389637, 4.17204937211163430221836374047, 5.32965192499966450627569329263, 6.03417713496676676718654486671, 6.32376885367540155077894281945, 7.07606662962461093646538640068, 7.86842127511508525968543300761