L(s) = 1 | + (−1.15 − 0.810i)2-s + (−1.50 + 0.861i)3-s + (0.686 + 1.87i)4-s + (−1.19 − 1.19i)5-s + (2.43 + 0.219i)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 + (−2.64 − 2.23i)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 + (−0.867 + 0.497i)3-s + (0.343 + 0.939i)4-s + (−0.536 − 0.536i)5-s + (0.995 + 0.0897i)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 + (−0.764 − 0.644i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0430 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0430 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.251985 + 0.263080i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.251985 + 0.263080i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.50 - 0.861i)T \) |
| 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
−11.68256424255957977853385347866, −10.69478215016910836222674512539, −10.25905359768570264172013006377, −9.035478476668041277715802620973, −8.265061084550581900554365636298, −7.20969351783786297116999846153, −5.93960783767797650070777619006, −4.55291497166183514109168449948, −3.67782259730462889070989365287, −1.56556970429272903465134209795,
0.37818485410838890889472947642, 2.33978871744282169123749905038, 4.58186840054548880324709904329, 5.68967965849388648272213401900, 6.68105606557913519942399910882, 7.44349638247713276621369074630, 8.221793015475550246519534340454, 9.418138125641123353067625715922, 10.65158224568642713811974543926, 11.12708415781181512888050927320