| L(s) = 1 | + 2-s − 1.44·3-s + 4-s + 2.24·5-s − 1.44·6-s + 7-s + 8-s − 0.911·9-s + 2.24·10-s − 2.19·11-s − 1.44·12-s + 14-s − 3.24·15-s + 16-s − 8.00·17-s − 0.911·18-s − 4.44·19-s + 2.24·20-s − 1.44·21-s − 2.19·22-s − 4.13·23-s − 1.44·24-s + 0.0489·25-s + 5.65·27-s + 28-s − 8.45·29-s − 3.24·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.834·3-s + 0.5·4-s + 1.00·5-s − 0.589·6-s + 0.377·7-s + 0.353·8-s − 0.303·9-s + 0.710·10-s − 0.662·11-s − 0.417·12-s + 0.267·14-s − 0.838·15-s + 0.250·16-s − 1.94·17-s − 0.214·18-s − 1.01·19-s + 0.502·20-s − 0.315·21-s − 0.468·22-s − 0.862·23-s − 0.294·24-s + 0.00978·25-s + 1.08·27-s + 0.188·28-s − 1.57·29-s − 0.592·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.44T + 3T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 17 | \( 1 + 8.00T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 + 4.13T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 + 8.93T + 43T^{2} \) |
| 47 | \( 1 + 0.472T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 + 0.0217T + 61T^{2} \) |
| 67 | \( 1 + 6.66T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 3.15T + 73T^{2} \) |
| 79 | \( 1 - 2.77T + 79T^{2} \) |
| 83 | \( 1 - 1.91T + 83T^{2} \) |
| 89 | \( 1 + 9.72T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
| 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
−8.623169130695825549352712311703, −7.68275851052213362877802942761, −6.65965050857867121791186081624, −6.07756625811421436202407550524, −5.51994628126480442821832616723, −4.77036009740738915570162944635, −3.93592221864542937161726711683, −2.45299773343321934348065534493, −1.92928701439909896915340697405, 0,
1.92928701439909896915340697405, 2.45299773343321934348065534493, 3.93592221864542937161726711683, 4.77036009740738915570162944635, 5.51994628126480442821832616723, 6.07756625811421436202407550524, 6.65965050857867121791186081624, 7.68275851052213362877802942761, 8.623169130695825549352712311703
