| L(s) = 1 | + (3.23 − 2.35i)2-s + (−0.839 + 2.58i)3-s + (4.94 − 15.2i)4-s + (−19.5 + 52.3i)5-s + (3.35 + 10.3i)6-s + 200.·7-s + (−19.7 − 60.8i)8-s + (190. + 138. i)9-s + (59.8 + 215. i)10-s + (273. − 198. i)11-s + (35.1 + 25.5i)12-s + (−211. − 153. i)13-s + (650. − 472. i)14-s + (−118. − 94.5i)15-s + (−207. − 150. i)16-s + (477. + 1.47e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.0538 + 0.165i)3-s + (0.154 − 0.475i)4-s + (−0.349 + 0.936i)5-s + (0.0380 + 0.117i)6-s + 1.55·7-s + (−0.109 − 0.336i)8-s + (0.784 + 0.569i)9-s + (0.189 + 0.681i)10-s + (0.681 − 0.495i)11-s + (0.0705 + 0.0512i)12-s + (−0.346 − 0.251i)13-s + (0.886 − 0.644i)14-s + (−0.136 − 0.108i)15-s + (−0.202 − 0.146i)16-s + (0.400 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0393i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.49954 + 0.0492272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.49954 + 0.0492272i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 5 | \( 1 + (19.5 - 52.3i)T \) |
| good | 3 | \( 1 + (0.839 - 2.58i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 - 200.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-273. + 198. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (211. + 153. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-477. - 1.47e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (201. + 618. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-994. + 722. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.35e3 - 4.17e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (123. + 379. i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (2.50e3 + 1.81e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.23e4 + 8.97e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.88e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.16e3 + 6.66e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.21e4 + 3.73e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.76e4 - 2.73e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.88e4 - 1.37e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.80e4 + 5.56e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.52e3 - 7.77e3i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-5.48e4 + 3.98e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (9.00e3 - 2.77e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (5.30e3 + 1.63e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (7.77e3 - 5.65e3i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.81e4 + 5.57e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−14.69307918647975449958255173112, −13.54167987811106394148662057695, −12.04976214114266182907111842065, −11.03409771078147366556013505494, −10.34089338348103454520110972803, −8.312073376970718087025725028989, −6.95547548778933554348502155883, −5.15051333439321267299748932920, −3.77443126591126499659225077297, −1.78033654004440991271047628526,
1.42528341569405308095357091168, 4.21068735889859841402621048694, 5.14776491772662267175364236470, 7.07521604797208857720358162611, 8.151577728753871259839503174511, 9.535902276608123682744095052837, 11.62894161168038278695441716671, 12.12090780973886378594059585267, 13.46027936180142267240047616564, 14.63038066578364095584369191949
