| L(s) = 1 | + 9·3-s + 37.1i·5-s − 176. i·7-s + 81·9-s + 179. i·11-s + (−554. − 252. i)13-s + 334. i·15-s − 933.·17-s + 2.33e3i·19-s − 1.58e3i·21-s + 2.79e3·23-s + 1.74e3·25-s + 729·27-s + 1.50e3·29-s + 737. i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.663i·5-s − 1.36i·7-s + 0.333·9-s + 0.447i·11-s + (−0.910 − 0.413i)13-s + 0.383i·15-s − 0.783·17-s + 1.48i·19-s − 0.785i·21-s + 1.10·23-s + 0.559·25-s + 0.192·27-s + 0.331·29-s + 0.137i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.520809341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520809341\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 13 | \( 1 + (554. + 252. i)T \) |
| good | 5 | \( 1 - 37.1iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 176. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 179. iT - 1.61e5T^{2} \) |
| 17 | \( 1 + 933.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.33e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 737. iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 3.77e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 368. iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.05e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.50e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.53e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.66e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.89e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 3.41e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.49e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.22e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.17e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 2.80e4iT - 8.58e9T^{2} \) |
| 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
−9.994321990751631726211909834170, −8.934646620525931159010058621146, −7.77719226677921782587794991850, −7.25478744652236884443160063417, −6.48934260075849633188225203430, −5.00046622963145585683443603479, −4.03331587516143746352458234440, −3.12890028179350542320648533678, −1.99689509381207852773075736137, −0.64610803080619483784410394620,
0.845292682097967014254869906180, 2.30475672189947775061111889174, 2.90709900838358671446565450491, 4.53171258024345447444663419704, 5.12671468352420220442452956881, 6.32710787222062382506275000394, 7.29833141133885001936709906848, 8.441244608034686328082594893188, 9.097536459028323040203458296051, 9.382260199361189844657791496096
