L(s) = 1 | + (−0.331 − 1.37i)2-s + 2.13i·3-s + (−1.78 + 0.910i)4-s + (−1.94 + 1.10i)5-s + (2.93 − 0.707i)6-s + (−2.35 + 1.19i)7-s + (1.84 + 2.14i)8-s − 1.56·9-s + (2.16 + 2.30i)10-s + 2.33i·11-s + (−1.94 − 3.80i)12-s − 1.09·13-s + (2.42 + 2.84i)14-s + (−2.35 − 4.15i)15-s + (2.34 − 3.24i)16-s + 4.98·17-s + ⋯ |
L(s) = 1 | + (−0.234 − 0.972i)2-s + 1.23i·3-s + (−0.890 + 0.455i)4-s + (−0.869 + 0.493i)5-s + (1.19 − 0.288i)6-s + (−0.891 + 0.453i)7-s + (0.650 + 0.759i)8-s − 0.520·9-s + (0.683 + 0.729i)10-s + 0.703i·11-s + (−0.561 − 1.09i)12-s − 0.302·13-s + (0.649 + 0.760i)14-s + (−0.608 − 1.07i)15-s + (0.585 − 0.810i)16-s + 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.470910 + 0.421231i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.470910 + 0.421231i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.331 + 1.37i)T \) |
| 5 | \( 1 + (1.94 - 1.10i)T \) |
| 7 | \( 1 + (2.35 - 1.19i)T \) |
good | 3 | \( 1 - 2.13iT - 3T^{2} \) |
| 11 | \( 1 - 2.33iT - 11T^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 - 0.561T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 - 5.49iT - 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 + 9.74iT - 47T^{2} \) |
| 53 | \( 1 - 8.58iT - 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 0.620iT - 61T^{2} \) |
| 67 | \( 1 - 4.71T + 67T^{2} \) |
| 71 | \( 1 - 11.9iT - 71T^{2} \) |
| 73 | \( 1 - 9.96T + 73T^{2} \) |
| 79 | \( 1 + 10.6iT - 79T^{2} \) |
| 83 | \( 1 + 3.86iT - 83T^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−13.01556343673776775285994559567, −12.20552193900367650321177600306, −11.21326894312536105579687813704, −10.22267743665441494726632292261, −9.664614169815255061692444923777, −8.573170597585043681542905694115, −7.15148823093855921930890825974, −5.11598769577960765906163377403, −3.91942580182936009648197182719, −2.98181506426816068131242571592,
0.72683490191407248632891211873, 3.72801957616220722941569000900, 5.45394186989794757216075183549, 6.76002990168634668467539449535, 7.44284703896194459525808874985, 8.347086015348734524507022827288, 9.457903643628146418726591451391, 10.89082067897187859257165783740, 12.46323764391199689203404359513, 12.86881337842059207864959504410