L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 6.41·11-s − 12-s + 13-s − 15-s + 16-s − 3.24·17-s + 18-s + 4.65·19-s + 20-s − 6.41·22-s − 2.17·23-s − 24-s − 4·25-s + 26-s − 27-s + 7.48·29-s − 30-s − 1.41·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.93·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 0.250·16-s − 0.786·17-s + 0.235·18-s + 1.06·19-s + 0.223·20-s − 1.36·22-s − 0.452·23-s − 0.204·24-s − 0.800·25-s + 0.196·26-s − 0.192·27-s + 1.38·29-s − 0.182·30-s − 0.254·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 6.41T + 11T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 8.41T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 5.75T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 2.24T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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
−7.920780859313957004596682522926, −7.32226953267307963064307580282, −6.41653785719251234976125057002, −5.78492248970426685553312608540, −5.10909179830418401912722999775, −4.60020722500704982587396409857, −3.39911339245620377301034545645, −2.59513001510396615719384651117, −1.61732694963993015620130702052, 0,
1.61732694963993015620130702052, 2.59513001510396615719384651117, 3.39911339245620377301034545645, 4.60020722500704982587396409857, 5.10909179830418401912722999775, 5.78492248970426685553312608540, 6.41653785719251234976125057002, 7.32226953267307963064307580282, 7.920780859313957004596682522926