| 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
−11.83939621558647634743520261579, −11.14844365909870459248737519289, −10.44005450636669484137582873090, −8.949600202750637811637843104501, −8.563553090257301648105886321529, −6.66588491079092384650466087521, −5.59858137560061121012524953462, −4.88274104105306138840273801223, −2.86125630459363451857361989845, −1.29674872469290547821606179248,
1.75692082009687344122246733528, 4.53603057523109126500313541934, 4.98632156035764521028656231098, 6.44525588631750228968243499026, 7.02798753804830576842731117453, 8.477043047126065774249212528338, 9.495565678357594522024747356583, 10.43384393784295576350638121620, 11.42131158038063383743027425733, 12.24207063875276352459485927228
