| L(s) = 1 | + (3.23 − 2.35i)2-s + (4.94 − 15.2i)4-s + (34.0 + 24.7i)5-s + (58.4 − 179. i)7-s + (−19.7 − 60.8i)8-s + 168.·10-s + (−393. − 76.3i)11-s + (193. − 140. i)13-s + (−233. − 719. i)14-s + (−207. − 150. i)16-s + (−9.75 − 7.08i)17-s + (1.49 + 4.59i)19-s + (544. − 395. i)20-s + (−1.45e3 + 679. i)22-s + 898.·23-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (0.608 + 0.442i)5-s + (0.450 − 1.38i)7-s + (−0.109 − 0.336i)8-s + 0.531·10-s + (−0.981 − 0.190i)11-s + (0.316 − 0.230i)13-s + (−0.318 − 0.980i)14-s + (−0.202 − 0.146i)16-s + (−0.00818 − 0.00594i)17-s + (0.000949 + 0.00292i)19-s + (0.304 − 0.221i)20-s + (−0.640 + 0.299i)22-s + 0.354·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.863i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.671867755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.671867755\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (393. + 76.3i)T \) |
| good | 5 | \( 1 + (-34.0 - 24.7i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-58.4 + 179. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-193. + 140. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (9.75 + 7.08i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.49 - 4.59i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 898.T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-623. + 1.91e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (95.7 - 69.5i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.97e3 + 1.22e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.91e3 + 1.20e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.12e3 - 6.54e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (5.95e3 - 4.32e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (2.33e3 - 7.19e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.96e4 + 1.42e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + 1.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-5.36e4 - 3.89e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.74e4 + 8.43e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.15e4 + 8.36e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-9.62e4 - 6.99e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 2.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.79e4 - 6.38e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
| 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
−10.91008050920807973286914444154, −10.66471243127921677257626702752, −9.659108520283950009301092898709, −8.078204115986168166590374737615, −7.05387190539099731769472863295, −5.87477717365584973336221490698, −4.70951373542822122229255109959, −3.50120292460975341427723674003, −2.14935106015318891672659812578, −0.64904600301461042964999284770,
1.77172516797547203022949453943, 2.98300778915398979550453282248, 4.84808237506246571490888575035, 5.45064565552126165562453014310, 6.50234714506122101846366493861, 7.976036041858473875823368661273, 8.764628087728467303531825187464, 9.811193613124636294530610670562, 11.19803630682194535527730186167, 12.09691305554728592483065841502
