| L(s) = 1 | + 36.0·2-s − 119.·3-s + 785.·4-s + 917.·5-s − 4.31e3·6-s − 4.19e3·7-s + 9.83e3·8-s − 5.29e3·9-s + 3.30e4·10-s + 5.60e4·11-s − 9.41e4·12-s + 1.94e3·13-s − 1.51e5·14-s − 1.10e5·15-s − 4.77e4·16-s − 1.90e5·18-s + 3.48e5·19-s + 7.20e5·20-s + 5.03e5·21-s + 2.01e6·22-s + 2.87e5·23-s − 1.17e6·24-s − 1.11e6·25-s + 7.00e4·26-s + 2.99e6·27-s − 3.29e6·28-s − 3.74e6·29-s + ⋯ |
| L(s) = 1 | + 1.59·2-s − 0.854·3-s + 1.53·4-s + 0.656·5-s − 1.36·6-s − 0.660·7-s + 0.849·8-s − 0.269·9-s + 1.04·10-s + 1.15·11-s − 1.31·12-s + 0.0188·13-s − 1.05·14-s − 0.561·15-s − 0.181·16-s − 0.428·18-s + 0.613·19-s + 1.00·20-s + 0.564·21-s + 1.83·22-s + 0.214·23-s − 0.725·24-s − 0.569·25-s + 0.0300·26-s + 1.08·27-s − 1.01·28-s − 0.983·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 36.0T + 512T^{2} \) |
| 3 | \( 1 + 119.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 917.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.19e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.60e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.94e3T + 1.06e10T^{2} \) |
| 19 | \( 1 - 3.48e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.87e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.74e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.22e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.92e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.96e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.94e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.12e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.42e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.77e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.42e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.54e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.56e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.39e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.29e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.33e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.00e9T + 7.60e17T^{2} \) |
| 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
−9.906288750824954987761407539446, −9.032416911776214299310511036797, −7.22781555629502333136381831575, −6.16761762388913510303203970020, −5.91662220312444422682510518633, −4.88325393278793524005891445482, −3.79767179920872745595342277884, −2.84114338781230525812807592857, −1.51106792076969099752194522012, 0,
1.51106792076969099752194522012, 2.84114338781230525812807592857, 3.79767179920872745595342277884, 4.88325393278793524005891445482, 5.91662220312444422682510518633, 6.16761762388913510303203970020, 7.22781555629502333136381831575, 9.032416911776214299310511036797, 9.906288750824954987761407539446
