L(s) = 1 | + (−0.5 − 0.866i)2-s + (−1.25 − 1.19i)3-s + (−0.499 + 0.866i)4-s + (−1.80 − 1.04i)5-s + (−0.403 + 1.68i)6-s + (1.78 + 1.95i)7-s + 0.999·8-s + (0.160 + 2.99i)9-s + 2.08i·10-s + 2.15·11-s + (1.66 − 0.492i)12-s + (−0.217 + 3.59i)13-s + (0.800 − 2.52i)14-s + (1.02 + 3.45i)15-s + (−0.5 − 0.866i)16-s + (0.278 − 0.482i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.725 − 0.687i)3-s + (−0.249 + 0.433i)4-s + (−0.806 − 0.465i)5-s + (−0.164 + 0.687i)6-s + (0.674 + 0.738i)7-s + 0.353·8-s + (0.0534 + 0.998i)9-s + 0.658i·10-s + 0.650·11-s + (0.479 − 0.142i)12-s + (−0.0602 + 0.998i)13-s + (0.213 − 0.673i)14-s + (0.264 + 0.892i)15-s + (−0.125 − 0.216i)16-s + (0.0675 − 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.798682 - 0.299127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.798682 - 0.299127i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (1.25 + 1.19i)T \) |
| 7 | \( 1 + (-1.78 - 1.95i)T \) |
| 13 | \( 1 + (0.217 - 3.59i)T \) |
good | 5 | \( 1 + (1.80 + 1.04i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 17 | \( 1 + (-0.278 + 0.482i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 + (1.79 - 1.03i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.10 - 3.52i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.21 + 5.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.20 + 4.15i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.532 + 0.307i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.33 - 7.50i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.507 - 0.292i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.68 + 3.85i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.35 + 0.782i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 8.20iT - 61T^{2} \) |
| 67 | \( 1 - 14.2iT - 67T^{2} \) |
| 71 | \( 1 + (-6.52 - 11.3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.198 + 0.344i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.73 + 9.93i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 + (-7.54 + 4.35i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.64 + 2.85i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.22893554464329018034190768693, −9.812378422243354850399099998793, −8.892659000034446752467182502057, −8.053348697472817994310548490367, −7.30983714036303483868183631799, −6.11671883222349481068701924672, −4.96138357416116074381241893435, −4.06652426512640392751413720114, −2.31400485484480868684304641013, −1.06419662724183416097970549118,
0.837579404261287557388183463347, 3.44169877707950288159648806507, 4.37380656096319260505042457167, 5.33238852718354035271681929608, 6.40547782246718690538158259993, 7.35404804607553370935502504491, 8.047210489416285476981528064053, 9.164919090747655222838042950305, 10.22273149938295557977266083546, 10.70918491119776828914105131285