| L(s) = 1 | + (5.80 + 6.70i)2-s + (13.1 + 8.42i)3-s + (−2.09 + 14.5i)4-s + (−138. − 63.1i)5-s + (19.6 + 136. i)6-s + (130. − 443. i)7-s + (367. − 236. i)8-s + (100. + 221. i)9-s + (−379. − 1.29e3i)10-s + (208. + 180. i)11-s + (−150. + 173. i)12-s + (3.35e3 − 986. i)13-s + (3.73e3 − 1.70e3i)14-s + (−1.28e3 − 1.99e3i)15-s + (4.62e3 + 1.35e3i)16-s + (3.86e3 − 555. i)17-s + ⋯ |
| L(s) = 1 | + (0.726 + 0.838i)2-s + (0.485 + 0.312i)3-s + (−0.0326 + 0.227i)4-s + (−1.10 − 0.505i)5-s + (0.0911 + 0.633i)6-s + (0.380 − 1.29i)7-s + (0.718 − 0.461i)8-s + (0.138 + 0.303i)9-s + (−0.379 − 1.29i)10-s + (0.156 + 0.135i)11-s + (−0.0868 + 0.100i)12-s + (1.52 − 0.449i)13-s + (1.36 − 0.621i)14-s + (−0.379 − 0.590i)15-s + (1.12 + 0.331i)16-s + (0.787 − 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.86037 - 0.283188i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.86037 - 0.283188i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.1 - 8.42i)T \) |
| 23 | \( 1 + (1.21e4 + 967. i)T \) |
| good | 2 | \( 1 + (-5.80 - 6.70i)T + (-9.10 + 63.3i)T^{2} \) |
| 5 | \( 1 + (138. + 63.1i)T + (1.02e4 + 1.18e4i)T^{2} \) |
| 7 | \( 1 + (-130. + 443. i)T + (-9.89e4 - 6.36e4i)T^{2} \) |
| 11 | \( 1 + (-208. - 180. i)T + (2.52e5 + 1.75e6i)T^{2} \) |
| 13 | \( 1 + (-3.35e3 + 986. i)T + (4.06e6 - 2.60e6i)T^{2} \) |
| 17 | \( 1 + (-3.86e3 + 555. i)T + (2.31e7 - 6.80e6i)T^{2} \) |
| 19 | \( 1 + (1.18e4 + 1.70e3i)T + (4.51e7 + 1.32e7i)T^{2} \) |
| 29 | \( 1 + (984. + 6.84e3i)T + (-5.70e8 + 1.67e8i)T^{2} \) |
| 31 | \( 1 + (-2.56e4 + 1.64e4i)T + (3.68e8 - 8.07e8i)T^{2} \) |
| 37 | \( 1 + (-7.82e4 + 3.57e4i)T + (1.68e9 - 1.93e9i)T^{2} \) |
| 41 | \( 1 + (2.72e4 - 5.96e4i)T + (-3.11e9 - 3.58e9i)T^{2} \) |
| 43 | \( 1 + (-6.76e3 + 1.05e4i)T + (-2.62e9 - 5.75e9i)T^{2} \) |
| 47 | \( 1 - 2.79e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (5.36e4 - 1.82e5i)T + (-1.86e10 - 1.19e10i)T^{2} \) |
| 59 | \( 1 + (-1.58e5 + 4.64e4i)T + (3.54e10 - 2.28e10i)T^{2} \) |
| 61 | \( 1 + (1.80e5 + 2.80e5i)T + (-2.14e10 + 4.68e10i)T^{2} \) |
| 67 | \( 1 + (4.02e5 - 3.48e5i)T + (1.28e10 - 8.95e10i)T^{2} \) |
| 71 | \( 1 + (-1.61e5 - 1.86e5i)T + (-1.82e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (9.55e4 - 6.64e5i)T + (-1.45e11 - 4.26e10i)T^{2} \) |
| 79 | \( 1 + (-5.80e4 - 1.97e5i)T + (-2.04e11 + 1.31e11i)T^{2} \) |
| 83 | \( 1 + (2.17e5 - 9.91e4i)T + (2.14e11 - 2.47e11i)T^{2} \) |
| 89 | \( 1 + (-2.46e5 + 3.83e5i)T + (-2.06e11 - 4.52e11i)T^{2} \) |
| 97 | \( 1 + (-6.03e5 - 2.75e5i)T + (5.45e11 + 6.29e11i)T^{2} \) |
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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
−13.70423183277465355302655782680, −12.83021300711166627081717523929, −11.19002012609654996530276683250, −10.19251095044441166649127723095, −8.302339017492231210982720367800, −7.64915644202419709514569479592, −6.17687021818993875921258562916, −4.35518009222415645436015423185, −3.96430233660623186910904801232, −0.937357936661508399940416267275,
1.83560032884487563074572030014, 3.20513552522512081626720818975, 4.22564325063758084832472585212, 6.15517943019478209662042852579, 7.947544360086326809094886458918, 8.636359240360085210634846026986, 10.66561356433937455021793641368, 11.68831998816304229794693528249, 12.21332600800179600854426522063, 13.39106355178874217747063995032
