| L(s) = 1 | + 8·2-s − 20.8·3-s + 64·4-s + 216.·5-s − 167.·6-s + 512·8-s − 1.75e3·9-s + 1.73e3·10-s + 2.58e3·11-s − 1.33e3·12-s + 8.92e3·13-s − 4.52e3·15-s + 4.09e3·16-s + 1.11e4·17-s − 1.40e4·18-s + 8.64e3·19-s + 1.38e4·20-s + 2.06e4·22-s − 6.60e4·23-s − 1.06e4·24-s − 3.11e4·25-s + 7.13e4·26-s + 8.22e4·27-s + 1.28e5·29-s − 3.62e4·30-s + 2.04e5·31-s + 3.27e4·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.446·3-s + 0.5·4-s + 0.775·5-s − 0.315·6-s + 0.353·8-s − 0.800·9-s + 0.548·10-s + 0.584·11-s − 0.223·12-s + 1.12·13-s − 0.346·15-s + 0.250·16-s + 0.548·17-s − 0.566·18-s + 0.289·19-s + 0.387·20-s + 0.413·22-s − 1.13·23-s − 0.157·24-s − 0.398·25-s + 0.796·26-s + 0.803·27-s + 0.980·29-s − 0.244·30-s + 1.23·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.230978825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.230978825\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 20.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 216.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 2.58e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.11e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 8.64e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.60e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.28e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.04e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.91e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.23e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.20e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.27e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.68e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.12e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.34e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.19e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.35e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.39e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.51e6T + 8.07e13T^{2} \) |
| 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
−12.47081475771959736760449757338, −11.59752989319826970808790626767, −10.61499856601316767402627559887, −9.372830083780980812941048813524, −7.984567752194569375783974300840, −6.18946242750792278272848347253, −5.86967758712021433427887910180, −4.27153298183021187708993559745, −2.74661911354849801807531505003, −1.12111117009806743020517413200,
1.12111117009806743020517413200, 2.74661911354849801807531505003, 4.27153298183021187708993559745, 5.86967758712021433427887910180, 6.18946242750792278272848347253, 7.984567752194569375783974300840, 9.372830083780980812941048813524, 10.61499856601316767402627559887, 11.59752989319826970808790626767, 12.47081475771959736760449757338
