L(s) = 1 | + (0.389 − 1.45i)2-s + (0.239 − 1.71i)3-s + (−0.232 − 0.133i)4-s + (1.06 + 1.06i)5-s + (−2.40 − 1.01i)6-s + (1.36 − 0.366i)7-s + (1.84 − 1.84i)8-s + (−2.88 − 0.820i)9-s + (1.96 − 1.13i)10-s + (3.97 + 1.06i)11-s + (−0.285 + 0.366i)12-s − 2.12i·14-s + (2.08 − 1.57i)15-s + (−2.23 − 3.86i)16-s + (−2.51 + 4.36i)17-s + (−2.31 + 3.87i)18-s + ⋯ |
L(s) = 1 | + (0.275 − 1.02i)2-s + (0.138 − 0.990i)3-s + (−0.116 − 0.0669i)4-s + (0.476 + 0.476i)5-s + (−0.980 − 0.415i)6-s + (0.516 − 0.138i)7-s + (0.652 − 0.652i)8-s + (−0.961 − 0.273i)9-s + (0.621 − 0.358i)10-s + (1.19 + 0.321i)11-s + (−0.0823 + 0.105i)12-s − 0.569i·14-s + (0.537 − 0.405i)15-s + (−0.558 − 0.966i)16-s + (−0.611 + 1.05i)17-s + (−0.546 + 0.913i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15239 - 1.82706i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15239 - 1.82706i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.239 + 1.71i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.389 + 1.45i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (-1.06 - 1.06i)T + 5iT^{2} \) |
| 7 | \( 1 + (-1.36 + 0.366i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.97 - 1.06i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (2.51 - 4.36i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 3.73i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.20 - 3.58i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.46 + 2.46i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.40 - 5.23i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.45 - 5.42i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.90 + 1.09i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.25 - 4.25i)T - 47iT^{2} \) |
| 53 | \( 1 + 0.779iT - 53T^{2} \) |
| 59 | \( 1 + (-0.779 - 2.90i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.73 + 1.53i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.90 + 0.779i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.901 - 0.901i)T + 73iT^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + (-2.90 - 2.90i)T + 83iT^{2} \) |
| 89 | \( 1 + (-9.01 - 2.41i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.437 - 1.63i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.02424217485087855106620399863, −9.946035753392157673335445250233, −8.932826513470751701884075680234, −7.86714404037757516688894648315, −6.79661003055380179864316385842, −6.30900541301870218014978682083, −4.62458216902202625421554139472, −3.43704573978074693373045962452, −2.23506878727769997502493573503, −1.43618569632707664073550371686,
1.95431385992676353423379905981, 3.76735155073012539095355093411, 4.83263945375793450183228143678, 5.52962680081699556765988719639, 6.39660911231693040118240907032, 7.54682735836462895220596664113, 8.627035299469366512178847415218, 9.180270976553190888662344545322, 10.18687149792950436721551675129, 11.25545142371110997773388260410