| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 24.7·5-s − 36·6-s − 215.·7-s − 64·8-s + 81·9-s + 98.8·10-s + 144·12-s + 706.·13-s + 862.·14-s − 222.·15-s + 256·16-s − 1.65e3·17-s − 324·18-s + 947.·19-s − 395.·20-s − 1.93e3·21-s + 2.98e3·23-s − 576·24-s − 2.51e3·25-s − 2.82e3·26-s + 729·27-s − 3.44e3·28-s + 1.01e3·29-s + 889.·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.441·5-s − 0.408·6-s − 1.66·7-s − 0.353·8-s + 0.333·9-s + 0.312·10-s + 0.288·12-s + 1.15·13-s + 1.17·14-s − 0.255·15-s + 0.250·16-s − 1.38·17-s − 0.235·18-s + 0.602·19-s − 0.220·20-s − 0.959·21-s + 1.17·23-s − 0.204·24-s − 0.804·25-s − 0.819·26-s + 0.192·27-s − 0.831·28-s + 0.224·29-s + 0.180·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 24.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 215.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 706.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.65e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 947.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.65e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.37e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.22e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.58e4T + 8.58e9T^{2} \) |
| 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
−9.203599566940882809255240730076, −8.525881993770960633762619991512, −7.56570888168730315264268114913, −6.64099109177642920475799629524, −6.08344144833380743474190043377, −4.36613215903989865028575652686, −3.34099259322430030374167716811, −2.63731289443483466601420700248, −1.11016998104568896817479625530, 0,
1.11016998104568896817479625530, 2.63731289443483466601420700248, 3.34099259322430030374167716811, 4.36613215903989865028575652686, 6.08344144833380743474190043377, 6.64099109177642920475799629524, 7.56570888168730315264268114913, 8.525881993770960633762619991512, 9.203599566940882809255240730076
