| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 67.3·5-s − 36·6-s − 165.·7-s − 64·8-s + 81·9-s − 269.·10-s − 178.·11-s + 144·12-s + 328.·13-s + 660.·14-s + 606.·15-s + 256·16-s + 347.·17-s − 324·18-s + 116.·19-s + 1.07e3·20-s − 1.48e3·21-s + 712.·22-s − 4.82e3·23-s − 576·24-s + 1.40e3·25-s − 1.31e3·26-s + 729·27-s − 2.64e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.20·5-s − 0.408·6-s − 1.27·7-s − 0.353·8-s + 0.333·9-s − 0.851·10-s − 0.443·11-s + 0.288·12-s + 0.539·13-s + 0.900·14-s + 0.695·15-s + 0.250·16-s + 0.291·17-s − 0.235·18-s + 0.0740·19-s + 0.602·20-s − 0.735·21-s + 0.313·22-s − 1.90·23-s − 0.204·24-s + 0.451·25-s − 0.381·26-s + 0.192·27-s − 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 5 | \( 1 - 67.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 165.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 178.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 328.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 347.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 116.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 491.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.94e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.91e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.77e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 4.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.13e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.05e4T + 8.58e9T^{2} \) |
| 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
−9.917797139393632385871158001016, −9.451631810568800733054480429259, −8.502476808717377713085879759552, −7.42485937718405995877056099673, −6.31523648186555332162863839210, −5.66943296123675154228423728571, −3.75092323333434892972592249107, −2.62643724614557346275004915496, −1.62530100509711008011055274520, 0,
1.62530100509711008011055274520, 2.62643724614557346275004915496, 3.75092323333434892972592249107, 5.66943296123675154228423728571, 6.31523648186555332162863839210, 7.42485937718405995877056099673, 8.502476808717377713085879759552, 9.451631810568800733054480429259, 9.917797139393632385871158001016
