| L(s) = 1 | + (1.36 + 0.355i)2-s + (−2.00 + 1.33i)3-s + (1.74 + 0.973i)4-s + (0.756 − 0.150i)5-s + (−3.21 + 1.11i)6-s + (−1.69 − 4.08i)7-s + (2.04 + 1.95i)8-s + (1.06 − 2.58i)9-s + (1.08 + 0.0629i)10-s + (0.290 − 0.434i)11-s + (−4.79 + 0.389i)12-s + (−1.79 − 0.357i)13-s + (−0.863 − 6.19i)14-s + (−1.31 + 1.31i)15-s + (2.10 + 3.40i)16-s + (−3.04 − 3.04i)17-s + ⋯ |
| L(s) = 1 | + (0.967 + 0.251i)2-s + (−1.15 + 0.772i)3-s + (0.873 + 0.486i)4-s + (0.338 − 0.0673i)5-s + (−1.31 + 0.456i)6-s + (−0.639 − 1.54i)7-s + (0.723 + 0.690i)8-s + (0.356 − 0.860i)9-s + (0.344 + 0.0199i)10-s + (0.0874 − 0.130i)11-s + (−1.38 + 0.112i)12-s + (−0.497 − 0.0990i)13-s + (−0.230 − 1.65i)14-s + (−0.339 + 0.339i)15-s + (0.526 + 0.850i)16-s + (−0.737 − 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02068 + 0.392111i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02068 + 0.392111i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.355i)T \) |
| good | 3 | \( 1 + (2.00 - 1.33i)T + (1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (-0.756 + 0.150i)T + (4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (1.69 + 4.08i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.290 + 0.434i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (1.79 + 0.357i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (3.04 + 3.04i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.26 - 6.37i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-7.32 - 3.03i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (0.690 + 1.03i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + 1.55iT - 31T^{2} \) |
| 37 | \( 1 + (-0.371 - 1.86i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-6.15 - 2.54i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (7.03 + 4.70i)T + (16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (-1.12 - 1.12i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.92 + 5.88i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-0.738 + 0.146i)T + (54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (3.34 - 2.23i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (-3.05 + 2.03i)T + (25.6 - 61.8i)T^{2} \) |
| 71 | \( 1 + (0.317 + 0.766i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-0.292 + 0.706i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-6.17 + 6.17i)T - 79iT^{2} \) |
| 83 | \( 1 + (0.663 - 3.33i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (-12.3 + 5.12i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 3.44iT - 97T^{2} \) |
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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
−15.16376323302741981442262714478, −13.84220553912474836075310286516, −13.02353011834271133922130363883, −11.66550377972900374149806616699, −10.73841762840256148142582499202, −9.826885067822951925905074516475, −7.40218563845317738173864616298, −6.24719113644976978351749895293, −4.98906940837302051624653240122, −3.78702965994423154817315833694,
2.44402663602045645050231113359, 5.03371314026327438599506360206, 6.12016791539360734967128966480, 6.86290745801859979184848055577, 9.173637647882533078861903851212, 10.82345167198018164140273128194, 11.77222470894239189471927492996, 12.63598938434172062844247008820, 13.19233217130317169369302375193, 14.86356115521818802820173581373
