| L(s) = 1 | − 8·2-s + 89.9·3-s + 64·4-s − 125·5-s − 719.·6-s − 9.99e2·7-s − 512·8-s + 5.89e3·9-s + 1.00e3·10-s − 361.·11-s + 5.75e3·12-s − 1.43e4·13-s + 7.99e3·14-s − 1.12e4·15-s + 4.09e3·16-s + 2.56e4·17-s − 4.71e4·18-s + 2.01e3·19-s − 8.00e3·20-s − 8.99e4·21-s + 2.88e3·22-s + 1.09e4·23-s − 4.60e4·24-s + 1.56e4·25-s + 1.14e5·26-s + 3.33e5·27-s − 6.39e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.92·3-s + 0.5·4-s − 0.447·5-s − 1.35·6-s − 1.10·7-s − 0.353·8-s + 2.69·9-s + 0.316·10-s − 0.0818·11-s + 0.961·12-s − 1.80·13-s + 0.779·14-s − 0.859·15-s + 0.250·16-s + 1.26·17-s − 1.90·18-s + 0.0672·19-s − 0.223·20-s − 2.11·21-s + 0.0578·22-s + 0.187·23-s − 0.679·24-s + 0.199·25-s + 1.27·26-s + 3.26·27-s − 0.550·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 5 | \( 1 + 125T \) |
| 31 | \( 1 - 2.97e4T \) |
| good | 3 | \( 1 - 89.9T + 2.18e3T^{2} \) |
| 7 | \( 1 + 9.99e2T + 8.23e5T^{2} \) |
| 11 | \( 1 + 361.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.43e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.56e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.01e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.09e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.06e5T + 1.72e10T^{2} \) |
| 37 | \( 1 + 1.92e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.58e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.21e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.39e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.29e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.40e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.75e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 9.16e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.39e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.79e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.97e6T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.759097367980535486316664947890, −9.165939313172543153062880554135, −8.051038338499544008781050633350, −7.53299091109901610709041975476, −6.65158669645026159602188393434, −4.68228606713178976823297585923, −3.22061661654237902133094386772, −2.88532114163353721088031956643, −1.55191405430195262947140911728, 0,
1.55191405430195262947140911728, 2.88532114163353721088031956643, 3.22061661654237902133094386772, 4.68228606713178976823297585923, 6.65158669645026159602188393434, 7.53299091109901610709041975476, 8.051038338499544008781050633350, 9.165939313172543153062880554135, 9.759097367980535486316664947890
