L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.44 − 2.49i)3-s + (−0.499 + 0.866i)4-s + (−1.57 + 1.58i)5-s − 2.88·6-s + (−0.391 + 2.61i)7-s + 0.999·8-s + (−2.64 − 4.58i)9-s + (2.16 + 0.568i)10-s + (0.914 + 3.18i)11-s + (1.44 + 2.49i)12-s + 6.76i·13-s + (2.46 − 0.969i)14-s + (1.69 + 6.21i)15-s + (−0.5 − 0.866i)16-s + (1.33 + 0.772i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.831 − 1.44i)3-s + (−0.249 + 0.433i)4-s + (−0.703 + 0.710i)5-s − 1.17·6-s + (−0.147 + 0.989i)7-s + 0.353·8-s + (−0.882 − 1.52i)9-s + (0.683 + 0.179i)10-s + (0.275 + 0.961i)11-s + (0.415 + 0.720i)12-s + 1.87i·13-s + (0.657 − 0.259i)14-s + (0.437 + 1.60i)15-s + (−0.125 − 0.216i)16-s + (0.324 + 0.187i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.000420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.000420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25662 + 0.000264117i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25662 + 0.000264117i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (1.57 - 1.58i)T \) |
| 7 | \( 1 + (0.391 - 2.61i)T \) |
| 11 | \( 1 + (-0.914 - 3.18i)T \) |
good | 3 | \( 1 + (-1.44 + 2.49i)T + (-1.5 - 2.59i)T^{2} \) |
| 13 | \( 1 - 6.76iT - 13T^{2} \) |
| 17 | \( 1 + (-1.33 - 0.772i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.445 + 0.771i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.17 + 3.56i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.79iT - 29T^{2} \) |
| 31 | \( 1 + (-0.849 - 0.490i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.14 - 4.70i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.03T + 41T^{2} \) |
| 43 | \( 1 - 0.302T + 43T^{2} \) |
| 47 | \( 1 + (-5.24 - 9.08i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.89 - 2.82i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.61 - 2.08i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.226 + 0.393i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.39 + 4.84i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.72T + 71T^{2} \) |
| 73 | \( 1 + (-2.80 - 1.61i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.09 + 4.09i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.01iT - 83T^{2} \) |
| 89 | \( 1 + (-6.83 + 3.94i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 19.1T + 97T^{2} \) |
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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
−10.31521101435484903864411648420, −9.007738209504576558879757671640, −8.853796478703037685702383721813, −7.73973377692692927765245904034, −6.91389291479148927053168035665, −6.54268684478942720648323234517, −4.63606902381350267641487084841, −3.33644936554168541293920488206, −2.44052069509704195617066162040, −1.60379262251853188393764298274,
0.68046908781728473225295443649, 3.31560101880407677192155034399, 3.72960208828036957034544505377, 4.93890900236435447131019608395, 5.55525970491311269340670571008, 7.20044115378542341173457171761, 8.018055938393303329953499774877, 8.604090502993027057811926040843, 9.296275649947990639207822458311, 10.32586128218112980429086691430