| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (−1.41 + 0.307i)3-s + (−0.989 − 0.142i)4-s + (2.23 + 0.123i)5-s + (−0.205 − 1.43i)6-s + (3.38 − 1.84i)7-s + (0.212 − 0.977i)8-s + (−0.829 + 0.378i)9-s + (−0.282 + 2.21i)10-s + (3.85 + 3.34i)11-s + (1.44 − 0.103i)12-s + (−2.15 + 3.94i)13-s + (1.59 + 3.50i)14-s + (−3.19 + 0.511i)15-s + (0.959 + 0.281i)16-s + (−3.52 − 2.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (−0.815 + 0.177i)3-s + (−0.494 − 0.0711i)4-s + (0.998 + 0.0550i)5-s + (−0.0839 − 0.583i)6-s + (1.27 − 0.697i)7-s + (0.0751 − 0.345i)8-s + (−0.276 + 0.126i)9-s + (−0.0892 + 0.701i)10-s + (1.16 + 1.00i)11-s + (0.416 − 0.0297i)12-s + (−0.597 + 1.09i)13-s + (0.427 + 0.936i)14-s + (−0.823 + 0.132i)15-s + (0.239 + 0.0704i)16-s + (−0.855 − 0.640i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.916942 + 0.665206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.916942 + 0.665206i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 5 | \( 1 + (-2.23 - 0.123i)T \) |
| 23 | \( 1 + (-4.16 + 2.37i)T \) |
| good | 3 | \( 1 + (1.41 - 0.307i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-3.38 + 1.84i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-3.85 - 3.34i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.15 - 3.94i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (3.52 + 2.64i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.596 - 4.15i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-5.29 + 0.761i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.10 + 1.35i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (10.5 + 3.94i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (2.05 - 4.50i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.746 - 3.43i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (4.84 + 4.84i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.38 + 8.03i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (0.493 + 1.68i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.855 + 1.33i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.98 - 0.499i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (9.09 + 10.4i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (2.70 + 3.61i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (4.81 - 1.41i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.01 + 5.41i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (0.368 + 0.237i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.19 - 16.6i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−12.24207063875276352459485927228, −11.42131158038063383743027425733, −10.43384393784295576350638121620, −9.495565678357594522024747356583, −8.477043047126065774249212528338, −7.02798753804830576842731117453, −6.44525588631750228968243499026, −4.98632156035764521028656231098, −4.53603057523109126500313541934, −1.75692082009687344122246733528,
1.29674872469290547821606179248, 2.86125630459363451857361989845, 4.88274104105306138840273801223, 5.59858137560061121012524953462, 6.66588491079092384650466087521, 8.563553090257301648105886321529, 8.949600202750637811637843104501, 10.44005450636669484137582873090, 11.14844365909870459248737519289, 11.83939621558647634743520261579
