| L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−1 + 1.73i)5-s − 7.99·8-s + (1.99 + 3.46i)10-s + (−4 − 6.92i)11-s + 42·13-s + (−8 + 13.8i)16-s + (1 + 1.73i)17-s + (−62 + 107. i)19-s + 7.99·20-s − 15.9·22-s + (38 − 65.8i)23-s + (60.5 + 104. i)25-s + (42 − 72.7i)26-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.0894 + 0.154i)5-s − 0.353·8-s + (0.0632 + 0.109i)10-s + (−0.109 − 0.189i)11-s + 0.896·13-s + (−0.125 + 0.216i)16-s + (0.0142 + 0.0247i)17-s + (−0.748 + 1.29i)19-s + 0.0894·20-s − 0.155·22-s + (0.344 − 0.596i)23-s + (0.483 + 0.838i)25-s + (0.316 − 0.548i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.984735503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984735503\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (1 - 1.73i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (4 + 6.92i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 42T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (62 - 107. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-38 + 65.8i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 254T + 2.43e4T^{2} \) |
| 31 | \( 1 + (36 + 62.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (199 - 344. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 462T + 6.89e4T^{2} \) |
| 43 | \( 1 - 212T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-132 + 228. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (81 + 140. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-386 - 668. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-15 + 25.9i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-382 - 661. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 236T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-209 - 361. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (276 - 478. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (15 - 25.9i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
| 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
−9.957270487107779598788557667174, −8.991737098161316152193443610037, −8.263589572738720090504946913389, −7.18035821806291398183355251904, −6.11204886251247609649582088476, −5.41098609773173592481697392778, −4.13927702464579576629370436936, −3.46669154521627160513788424378, −2.22865774881789327106494750081, −1.05762368893188472857780654928,
0.52592008007404954291950668128, 2.19487981903011731595642355015, 3.51874473108059713856634664889, 4.41148650757609524056840775089, 5.37599940225723945169086226054, 6.23761548052942436531927234289, 7.12955943909573406754602001828, 7.88763349671280013482254588422, 8.931601117430705061972559435484, 9.335623285948798472617202246710
