| L(s) = 1 | + (0.230 + 0.0120i)2-s + (6.35 + 7.84i)3-s + (−7.90 − 0.830i)4-s + (9.14 − 6.42i)5-s + (1.36 + 1.88i)6-s + (14.8 + 11.0i)7-s + (−3.63 − 0.575i)8-s + (−15.5 + 73.3i)9-s + (2.18 − 1.36i)10-s + (−30.1 + 6.40i)11-s + (−43.7 − 67.3i)12-s + (39.6 + 20.2i)13-s + (3.28 + 2.73i)14-s + (108. + 30.9i)15-s + (61.3 + 13.0i)16-s + (−23.6 − 61.4i)17-s + ⋯ |
| L(s) = 1 | + (0.0813 + 0.00426i)2-s + (1.22 + 1.51i)3-s + (−0.987 − 0.103i)4-s + (0.818 − 0.575i)5-s + (0.0930 + 0.128i)6-s + (0.801 + 0.598i)7-s + (−0.160 − 0.0254i)8-s + (−0.577 + 2.71i)9-s + (0.0690 − 0.0433i)10-s + (−0.825 + 0.175i)11-s + (−1.05 − 1.61i)12-s + (0.845 + 0.431i)13-s + (0.0626 + 0.0521i)14-s + (1.86 + 0.532i)15-s + (0.958 + 0.203i)16-s + (−0.336 − 0.877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.183 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.56709 + 1.88755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56709 + 1.88755i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-9.14 + 6.42i)T \) |
| 7 | \( 1 + (-14.8 - 11.0i)T \) |
| good | 2 | \( 1 + (-0.230 - 0.0120i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (-6.35 - 7.84i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (30.1 - 6.40i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-39.6 - 20.2i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (23.6 + 61.4i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-0.806 - 7.67i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (4.70 - 89.8i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (80.6 - 110. i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-28.6 + 64.3i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (-9.88 + 6.42i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (-60.6 + 19.6i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (179. + 179. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (61.7 + 23.6i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (-494. + 400. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (-32.2 + 35.8i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-539. + 485. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (-870. + 334. i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-663. - 481. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-573. + 882. i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (-60.3 - 135. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (-74.9 + 473. i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (712. + 791. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (40.3 + 254. i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−13.00474300185756091876008448575, −11.24679335044902342613729932270, −10.12180191430162074077723839738, −9.375861895655212551460249562984, −8.754049725336695059477943471054, −8.048527640192367537859413955712, −5.34771706550116451563354158001, −4.95716168550239782445649668723, −3.72503430188512512783422327999, −2.15635115284315395791236502234,
1.06368100613493993662943829206, 2.46684127015152994724996782479, 3.80431498962464488614268317916, 5.71854915594240041138618825599, 6.90576932518495189248406243550, 8.122708402443772609680378808972, 8.472941007584600804862781709604, 9.764242945550491825945769084294, 10.94876361095732128275929655220, 12.52243540707160588388467882004
