| 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 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.335623285948798472617202246710, −8.931601117430705061972559435484, −7.88763349671280013482254588422, −7.12955943909573406754602001828, −6.23761548052942436531927234289, −5.37599940225723945169086226054, −4.41148650757609524056840775089, −3.51874473108059713856634664889, −2.19487981903011731595642355015, −0.52592008007404954291950668128,
1.05762368893188472857780654928, 2.22865774881789327106494750081, 3.46669154521627160513788424378, 4.13927702464579576629370436936, 5.41098609773173592481697392778, 6.11204886251247609649582088476, 7.18035821806291398183355251904, 8.263589572738720090504946913389, 8.991737098161316152193443610037, 9.957270487107779598788557667174
