| L(s) = 1 | + (−0.227 + 1.71i)3-s − 5-s + (−1.22 − 2.34i)7-s + (−2.89 − 0.781i)9-s + 4.73i·11-s − 5.25i·13-s + (0.227 − 1.71i)15-s + 0.432·17-s − 6.30i·19-s + (4.30 − 1.56i)21-s + 0.332i·23-s + 25-s + (2.00 − 4.79i)27-s − 7.48i·29-s − 0.0758i·31-s + ⋯ |
| L(s) = 1 | + (−0.131 + 0.991i)3-s − 0.447·5-s + (−0.461 − 0.887i)7-s + (−0.965 − 0.260i)9-s + 1.42i·11-s − 1.45i·13-s + (0.0587 − 0.443i)15-s + 0.104·17-s − 1.44i·19-s + (0.940 − 0.340i)21-s + 0.0692i·23-s + 0.200·25-s + (0.385 − 0.922i)27-s − 1.39i·29-s − 0.0136i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.624091 - 0.437584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.624091 - 0.437584i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.227 - 1.71i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (1.22 + 2.34i)T \) |
| good | 11 | \( 1 - 4.73iT - 11T^{2} \) |
| 13 | \( 1 + 5.25iT - 13T^{2} \) |
| 17 | \( 1 - 0.432T + 17T^{2} \) |
| 19 | \( 1 + 6.30iT - 19T^{2} \) |
| 23 | \( 1 - 0.332iT - 23T^{2} \) |
| 29 | \( 1 + 7.48iT - 29T^{2} \) |
| 31 | \( 1 + 0.0758iT - 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 - 1.97T + 43T^{2} \) |
| 47 | \( 1 - 9.28T + 47T^{2} \) |
| 53 | \( 1 + 13.8iT - 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 5.11iT - 61T^{2} \) |
| 67 | \( 1 - 6.75T + 67T^{2} \) |
| 71 | \( 1 - 9.35iT - 71T^{2} \) |
| 73 | \( 1 + 2.75iT - 73T^{2} \) |
| 79 | \( 1 - 0.0508T + 79T^{2} \) |
| 83 | \( 1 - 4.12T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−10.02278914733658797890103766711, −9.508079579152799531925485986946, −8.357327999556646576948280692358, −7.47962362156145399496183290499, −6.65265738581809805224699708894, −5.37969986232507492939809537648, −4.55832209228267404384179871976, −3.74681784150146819797263116342, −2.68707936948050471281288720202, −0.38795809214759296185524800454,
1.46221662182827388756188409947, 2.79014808527691718620720427124, 3.79984683999888403227597371554, 5.36999342353919551020662877214, 6.10421639166490800091241363390, 6.82811534417320639126991066623, 7.84671319264885741403525334092, 8.693792443396182053175093192542, 9.152771236887339371481678755153, 10.58985148501001489186468817713
