| L(s) = 1 | + (−0.647 + 0.250i)2-s + (−0.216 − 1.71i)3-s + (−1.12 + 1.02i)4-s + (0.349 − 1.22i)5-s + (0.571 + 1.05i)6-s + (−1.94 + 2.92i)7-s + (1.08 − 2.18i)8-s + (−2.90 + 0.743i)9-s + (0.0819 + 0.884i)10-s + (−3.24 − 2.61i)11-s + (1.99 + 1.70i)12-s + (−0.757 + 1.14i)13-s + (0.522 − 2.38i)14-s + (−2.18 − 0.335i)15-s + (0.123 − 1.33i)16-s + (5.28 − 2.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.458 + 0.177i)2-s + (−0.124 − 0.992i)3-s + (−0.560 + 0.511i)4-s + (0.156 − 0.549i)5-s + (0.233 + 0.432i)6-s + (−0.733 + 1.10i)7-s + (0.385 − 0.773i)8-s + (−0.968 + 0.247i)9-s + (0.0259 + 0.279i)10-s + (−0.978 − 0.787i)11-s + (0.577 + 0.492i)12-s + (−0.210 + 0.316i)13-s + (0.139 − 0.637i)14-s + (−0.565 − 0.0865i)15-s + (0.0308 − 0.332i)16-s + (1.28 − 0.687i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.733732 + 0.130144i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.733732 + 0.130144i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.216 + 1.71i)T \) |
| 103 | \( 1 + (4.59 - 9.04i)T \) |
| good | 2 | \( 1 + (0.647 - 0.250i)T + (1.47 - 1.34i)T^{2} \) |
| 5 | \( 1 + (-0.349 + 1.22i)T + (-4.25 - 2.63i)T^{2} \) |
| 7 | \( 1 + (1.94 - 2.92i)T + (-2.72 - 6.44i)T^{2} \) |
| 11 | \( 1 + (3.24 + 2.61i)T + (2.35 + 10.7i)T^{2} \) |
| 13 | \( 1 + (0.757 - 1.14i)T + (-5.06 - 11.9i)T^{2} \) |
| 17 | \( 1 + (-5.28 + 2.83i)T + (9.39 - 14.1i)T^{2} \) |
| 19 | \( 1 + (-1.78 - 5.06i)T + (-14.8 + 11.9i)T^{2} \) |
| 23 | \( 1 + (-0.386 - 2.49i)T + (-21.9 + 6.97i)T^{2} \) |
| 29 | \( 1 + (-0.601 + 2.11i)T + (-24.6 - 15.2i)T^{2} \) |
| 31 | \( 1 + (3.23 - 1.48i)T + (20.1 - 23.5i)T^{2} \) |
| 37 | \( 1 + (-4.68 + 6.20i)T + (-10.1 - 35.5i)T^{2} \) |
| 41 | \( 1 + (2.13 + 0.536i)T + (36.1 + 19.3i)T^{2} \) |
| 43 | \( 1 + (-3.45 - 4.58i)T + (-11.7 + 41.3i)T^{2} \) |
| 47 | \( 1 + (-1.90 - 3.29i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.470 + 0.548i)T + (-8.12 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-2.42 - 3.66i)T + (-22.9 + 54.3i)T^{2} \) |
| 61 | \( 1 + (-8.23 + 4.41i)T + (33.6 - 50.8i)T^{2} \) |
| 67 | \( 1 + (-3.30 + 6.63i)T + (-40.3 - 53.4i)T^{2} \) |
| 71 | \( 1 + (-15.4 - 3.88i)T + (62.5 + 33.5i)T^{2} \) |
| 73 | \( 1 + (-2.53 - 8.91i)T + (-62.0 + 38.4i)T^{2} \) |
| 79 | \( 1 + (-6.54 - 6.75i)T + (-2.43 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-2.02 - 4.06i)T + (-50.0 + 66.2i)T^{2} \) |
| 89 | \( 1 + (0.485 + 0.442i)T + (8.21 + 88.6i)T^{2} \) |
| 97 | \( 1 + (3.39 - 2.10i)T + (43.2 - 86.8i)T^{2} \) |
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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
−9.647306486823562798441557574986, −9.286532417786113733617716667288, −8.234524092310758628278970990827, −7.88220558142783791533457309428, −6.87772753459473328546798806281, −5.62665007544232135380963410384, −5.34064584143207897445887563500, −3.51812808555684822279107101726, −2.59644337250933549294884689427, −0.911300360037903870874042268152,
0.58790253235051945381516412818, 2.63451171355269745026142183879, 3.70795232200732470986204008500, 4.76086709210255443613530109882, 5.44772841327015634345854249371, 6.60711867521715766742619525509, 7.60023640333744410167061733186, 8.548515570667657286947213905703, 9.641992758864089940389272227988, 10.03039872874892818125145889714
