| L(s) = 1 | + (−3.64 − 7.57i)2-s + (2.80 − 5.82i)3-s + (−4.15 + 5.20i)4-s + (190. + 43.5i)5-s − 54.3·6-s + 593. i·7-s + (−469. − 107. i)8-s + (428. + 537. i)9-s + (−366. − 1.60e3i)10-s + (1.30e3 + 1.64e3i)11-s + (18.6 + 38.7i)12-s + (203. − 891. i)13-s + (4.49e3 − 2.16e3i)14-s + (789. − 989. i)15-s + (996. + 4.36e3i)16-s + (−837. − 3.67e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.455 − 0.946i)2-s + (0.103 − 0.215i)3-s + (−0.0648 + 0.0813i)4-s + (1.52 + 0.348i)5-s − 0.251·6-s + 1.73i·7-s + (−0.917 − 0.209i)8-s + (0.587 + 0.736i)9-s + (−0.366 − 1.60i)10-s + (0.983 + 1.23i)11-s + (0.0108 + 0.0224i)12-s + (0.0925 − 0.405i)13-s + (1.63 − 0.788i)14-s + (0.233 − 0.293i)15-s + (0.243 + 1.06i)16-s + (−0.170 − 0.747i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.85128 - 0.417780i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85128 - 0.417780i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-7.92e4 + 6.66e3i)T \) |
| good | 2 | \( 1 + (3.64 + 7.57i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (-2.80 + 5.82i)T + (-454. - 569. i)T^{2} \) |
| 5 | \( 1 + (-190. - 43.5i)T + (1.40e4 + 6.77e3i)T^{2} \) |
| 7 | \( 1 - 593. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + (-1.30e3 - 1.64e3i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-203. + 891. i)T + (-4.34e6 - 2.09e6i)T^{2} \) |
| 17 | \( 1 + (837. + 3.67e3i)T + (-2.17e7 + 1.04e7i)T^{2} \) |
| 19 | \( 1 + (6.30e3 + 5.02e3i)T + (1.04e7 + 4.58e7i)T^{2} \) |
| 23 | \( 1 + (-5.37e3 - 6.74e3i)T + (-3.29e7 + 1.44e8i)T^{2} \) |
| 29 | \( 1 + (6.98e3 + 1.45e4i)T + (-3.70e8 + 4.65e8i)T^{2} \) |
| 31 | \( 1 + (2.43e4 - 1.17e4i)T + (5.53e8 - 6.93e8i)T^{2} \) |
| 37 | \( 1 + 5.64e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (3.50e4 - 1.68e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-6.28e4 + 7.88e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (1.65e4 + 7.23e4i)T + (-1.99e10 + 9.61e9i)T^{2} \) |
| 59 | \( 1 + (-2.83e4 - 1.24e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-7.63e4 + 1.58e5i)T + (-3.21e10 - 4.02e10i)T^{2} \) |
| 67 | \( 1 + (-2.47e5 + 3.09e5i)T + (-2.01e10 - 8.81e10i)T^{2} \) |
| 71 | \( 1 + (9.47e4 + 7.55e4i)T + (2.85e10 + 1.24e11i)T^{2} \) |
| 73 | \( 1 + (-2.12e5 - 4.85e4i)T + (1.36e11 + 6.56e10i)T^{2} \) |
| 79 | \( 1 - 7.29e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (6.97e5 + 3.35e5i)T + (2.03e11 + 2.55e11i)T^{2} \) |
| 89 | \( 1 + (-2.05e5 + 4.26e5i)T + (-3.09e11 - 3.88e11i)T^{2} \) |
| 97 | \( 1 + (-6.45e4 - 8.09e4i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.62940979238380600577518878469, −13.10729300785313472868691295258, −12.19981293519425662649708055824, −10.86829272361135462397136923161, −9.618141456042957459438673718933, −9.089048754366800060740909597987, −6.70764737760299918733008172427, −5.41483479835542396049522724107, −2.39362383600653989270502951689, −1.86537398798419305695285146703,
1.15108749527241510102400396942, 3.87750379262590033851695313101, 6.10080400086208685592291007900, 6.84880783564042798483441169982, 8.592381963029098992806808215917, 9.594062437129235325270249287070, 10.79733196469870624446236881048, 12.74299101541609361450650133134, 13.91055526931863327513281713197, 14.69969534557075947328746073467
