| L(s) = 1 | + (2.28 − 0.670i)2-s + (−0.654 − 0.755i)3-s + (3.07 − 1.97i)4-s + (0.142 + 0.989i)5-s + (−2.00 − 1.28i)6-s + (0.858 − 1.87i)7-s + (2.58 − 2.97i)8-s + (−0.142 + 0.989i)9-s + (0.988 + 2.16i)10-s + (1.40 + 0.412i)11-s + (−3.50 − 1.03i)12-s + (−1.86 − 4.07i)13-s + (0.699 − 4.86i)14-s + (0.654 − 0.755i)15-s + (0.855 − 1.87i)16-s + (4.67 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + (1.61 − 0.473i)2-s + (−0.378 − 0.436i)3-s + (1.53 − 0.988i)4-s + (0.0636 + 0.442i)5-s + (−0.816 − 0.524i)6-s + (0.324 − 0.710i)7-s + (0.912 − 1.05i)8-s + (−0.0474 + 0.329i)9-s + (0.312 + 0.684i)10-s + (0.423 + 0.124i)11-s + (−1.01 − 0.297i)12-s + (−0.516 − 1.13i)13-s + (0.186 − 1.29i)14-s + (0.169 − 0.195i)15-s + (0.213 − 0.468i)16-s + (1.13 + 0.728i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.45101 - 1.46170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.45101 - 1.46170i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (4.78 + 0.253i)T \) |
| good | 2 | \( 1 + (-2.28 + 0.670i)T + (1.68 - 1.08i)T^{2} \) |
| 7 | \( 1 + (-0.858 + 1.87i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.40 - 0.412i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.86 + 4.07i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.67 - 3.00i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (4.30 - 2.76i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.35 - 2.79i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (4.95 - 5.71i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.214 - 1.49i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.992 + 6.90i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-4.25 - 4.91i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 + (-3.14 + 6.89i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (2.55 + 5.60i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (2.21 - 2.55i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-8.24 + 2.41i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (4.47 - 1.31i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (7.40 - 4.76i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.75 + 8.21i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.311 + 2.16i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (0.889 + 1.02i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.29 + 15.9i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.64486178711735456102411770340, −10.54046557293302894459685787209, −10.29348677823476807519272922853, −8.223244921570560715627742672352, −7.18734998946924603937978393322, −6.17452144977419950059670034019, −5.38325554831188641104762044807, −4.20278419542830813267318862268, −3.19977125430383836327635602307, −1.71144483064278725346354061254,
2.40465137320163837097363031890, 3.96637240660417343601287399703, 4.71828463047835028521500228366, 5.62202611337251349134189436419, 6.40151087002793750819579801099, 7.53808646409442689441800582492, 8.880473682726998924883271855413, 9.815955273815101620733691588821, 11.32336129738382788215218517303, 11.93052522273741091815432582439
