| L(s) = 1 | + (−0.178 − 0.780i)2-s + (−0.711 + 0.219i)3-s + (1.22 − 0.589i)4-s + (0.511 − 1.30i)5-s + (0.298 + 0.516i)6-s + (−2.37 + 4.12i)7-s + (−1.67 − 2.10i)8-s + (−2.02 + 1.37i)9-s + (−1.10 − 0.167i)10-s + (2.39 + 1.15i)11-s + (−0.741 + 0.688i)12-s + (0.611 − 0.0920i)13-s + (3.64 + 1.12i)14-s + (−0.0779 + 1.04i)15-s + (0.352 − 0.441i)16-s + (−2.15 − 5.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.125 − 0.551i)2-s + (−0.410 + 0.126i)3-s + (0.612 − 0.294i)4-s + (0.228 − 0.583i)5-s + (0.121 + 0.210i)6-s + (−0.899 + 1.55i)7-s + (−0.592 − 0.743i)8-s + (−0.673 + 0.459i)9-s + (−0.350 − 0.0528i)10-s + (0.722 + 0.348i)11-s + (−0.214 + 0.198i)12-s + (0.169 − 0.0255i)13-s + (0.972 + 0.300i)14-s + (−0.0201 + 0.268i)15-s + (0.0880 − 0.110i)16-s + (−0.523 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.685458 - 0.238697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.685458 - 0.238697i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.54 - 0.452i)T \) |
| good | 2 | \( 1 + (0.178 + 0.780i)T + (-1.80 + 0.867i)T^{2} \) |
| 3 | \( 1 + (0.711 - 0.219i)T + (2.47 - 1.68i)T^{2} \) |
| 5 | \( 1 + (-0.511 + 1.30i)T + (-3.66 - 3.40i)T^{2} \) |
| 7 | \( 1 + (2.37 - 4.12i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.39 - 1.15i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-0.611 + 0.0920i)T + (12.4 - 3.83i)T^{2} \) |
| 17 | \( 1 + (2.15 + 5.50i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (1.48 + 1.01i)T + (6.94 + 17.6i)T^{2} \) |
| 23 | \( 1 + (-0.178 - 2.37i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (1.79 + 0.555i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-5.49 + 5.09i)T + (2.31 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-1.11 - 1.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.583 + 2.55i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (8.07 - 3.88i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (-13.5 - 2.04i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (4.23 - 5.30i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-3.03 - 2.81i)T + (4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-4.65 - 3.17i)T + (24.4 + 62.3i)T^{2} \) |
| 71 | \( 1 + (0.641 - 8.55i)T + (-70.2 - 10.5i)T^{2} \) |
| 73 | \( 1 + (-8.41 + 1.26i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-3.09 + 5.36i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.21 - 0.374i)T + (68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (9.04 - 2.79i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + (-4.06 - 1.95i)T + (60.4 + 75.8i)T^{2} \) |
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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
−15.92079963806934145810443742270, −15.03088912992116517147559937373, −13.24519132386054470830692184896, −12.00008692563994333967588770877, −11.43002280173021349688388947783, −9.755061056819501252047637978047, −8.902409583769967728086372882141, −6.53610386599032064408175257162, −5.39183532714152129797005776017, −2.60514349455380226912408488264,
3.50821162799008248289469189440, 6.40849836052162299778787954914, 6.69043799420212312690847975563, 8.452370757451733455913983724659, 10.32081307579905870285247966831, 11.23537136076974712007017723714, 12.62413883811122254554828393666, 14.02441639465546991598716973660, 15.02267759101327636140706981338, 16.55207144978960255227524872223
