L(s) = 1 | + (1.15 − 0.810i)2-s + (−0.861 − 1.50i)3-s + (0.686 − 1.87i)4-s + (1.19 − 1.19i)5-s + (−2.21 − 1.04i)6-s + 7-s + (−0.726 − 2.73i)8-s + (−1.51 + 2.58i)9-s + (0.417 − 2.36i)10-s + (1.10 + 1.10i)11-s + (−3.41 + 0.585i)12-s + (0.418 − 0.418i)13-s + (1.15 − 0.810i)14-s + (−2.83 − 0.769i)15-s + (−3.05 − 2.57i)16-s + 4.01i·17-s + ⋯ |
L(s) = 1 | + (0.819 − 0.573i)2-s + (−0.497 − 0.867i)3-s + (0.343 − 0.939i)4-s + (0.536 − 0.536i)5-s + (−0.904 − 0.426i)6-s + 0.377·7-s + (−0.256 − 0.966i)8-s + (−0.505 + 0.862i)9-s + (0.132 − 0.746i)10-s + (0.332 + 0.332i)11-s + (−0.985 + 0.169i)12-s + (0.116 − 0.116i)13-s + (0.309 − 0.216i)14-s + (−0.731 − 0.198i)15-s + (−0.764 − 0.644i)16-s + 0.974i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.917649 - 1.68456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.917649 - 1.68456i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.15 + 0.810i)T \) |
| 3 | \( 1 + (0.861 + 1.50i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-1.19 + 1.19i)T - 5iT^{2} \) |
| 11 | \( 1 + (-1.10 - 1.10i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.418 + 0.418i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.01iT - 17T^{2} \) |
| 19 | \( 1 + (4.91 + 4.91i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.26iT - 23T^{2} \) |
| 29 | \( 1 + (-4.61 - 4.61i)T + 29iT^{2} \) |
| 31 | \( 1 + 7.31iT - 31T^{2} \) |
| 37 | \( 1 + (-2.61 - 2.61i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.46T + 41T^{2} \) |
| 43 | \( 1 + (-3.91 + 3.91i)T - 43iT^{2} \) |
| 47 | \( 1 + 8.08T + 47T^{2} \) |
| 53 | \( 1 + (-1.78 + 1.78i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.55 - 5.55i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.325 + 0.325i)T - 61iT^{2} \) |
| 67 | \( 1 + (1.41 + 1.41i)T + 67iT^{2} \) |
| 71 | \( 1 - 11.7iT - 71T^{2} \) |
| 73 | \( 1 + 2.89iT - 73T^{2} \) |
| 79 | \( 1 - 15.6iT - 79T^{2} \) |
| 83 | \( 1 + (-9.34 + 9.34i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + 3.43T + 97T^{2} \) |
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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
−11.34504109565002219529010379850, −10.73965752890424333024719135490, −9.545550440483334380298204903972, −8.429382974752674059533799505618, −7.07464012441292740174511464599, −6.14301980611987136990405225131, −5.29796184581437117692864235115, −4.25293574339324726883545896333, −2.39813805389776806613832433238, −1.27781966937305985350655390375,
2.66446802941304209950418796561, 3.99585520348730720055707396589, 4.90031341107422873241840578057, 6.06647375158127408932421170203, 6.56534107935890228531377839557, 8.033335735676257864764783457924, 9.035238166013345410250355479365, 10.23279321693109722778721618420, 11.01338879081392692003581543181, 11.87558351368360007187462863059