| L(s) = 1 | + (0.726 − 1.57i)3-s + (−2.20 + 0.341i)5-s + (2.06 + 1.64i)7-s + (−1.94 − 2.28i)9-s − 1.06i·11-s − 4.82·13-s + (−1.06 + 3.72i)15-s − 7.89i·17-s − 4.02i·19-s + (4.09 − 2.05i)21-s − 5.69·23-s + (4.76 − 1.50i)25-s + (−5.00 + 1.40i)27-s + 2.00i·29-s − 4.89i·31-s + ⋯ |
| L(s) = 1 | + (0.419 − 0.907i)3-s + (−0.988 + 0.152i)5-s + (0.781 + 0.623i)7-s + (−0.648 − 0.761i)9-s − 0.320i·11-s − 1.33·13-s + (−0.275 + 0.961i)15-s − 1.91i·17-s − 0.922i·19-s + (0.893 − 0.448i)21-s − 1.18·23-s + (0.953 − 0.301i)25-s + (−0.962 + 0.269i)27-s + 0.372i·29-s − 0.879i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.291701 - 0.913373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.291701 - 0.913373i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.726 + 1.57i)T \) |
| 5 | \( 1 + (2.20 - 0.341i)T \) |
| 7 | \( 1 + (-2.06 - 1.64i)T \) |
| good | 11 | \( 1 + 1.06iT - 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 7.89iT - 17T^{2} \) |
| 19 | \( 1 + 4.02iT - 19T^{2} \) |
| 23 | \( 1 + 5.69T + 23T^{2} \) |
| 29 | \( 1 - 2.00iT - 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 2.56iT - 37T^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 + 6.15iT - 43T^{2} \) |
| 47 | \( 1 + 2.27iT - 47T^{2} \) |
| 53 | \( 1 + 9.84T + 53T^{2} \) |
| 59 | \( 1 + 5.87T + 59T^{2} \) |
| 61 | \( 1 + 7.02iT - 61T^{2} \) |
| 67 | \( 1 - 10.8iT - 67T^{2} \) |
| 71 | \( 1 + 0.0512iT - 71T^{2} \) |
| 73 | \( 1 + 2.86T + 73T^{2} \) |
| 79 | \( 1 - 7.00T + 79T^{2} \) |
| 83 | \( 1 - 7.59iT - 83T^{2} \) |
| 89 | \( 1 - 9.72T + 89T^{2} \) |
| 97 | \( 1 - 1.77T + 97T^{2} \) |
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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
−9.594734727285212608449445488080, −8.902304497543433943689289792901, −7.923909077888802272569467998672, −7.50113872915297193222650954062, −6.67986350570290206540874557694, −5.37505628636108835698397266325, −4.49279247931031148997989986506, −3.02899765741170982295551111134, −2.26045478678182664899001583189, −0.42752549316358731413178009087,
1.89970850547553010007394321760, 3.44952554294035462819613450135, 4.26753880331339707844118907719, 4.81097806284835238467104496626, 6.08938291991318706562636289909, 7.66574596288175472598822996208, 7.85440937672740818802078151160, 8.747656708355441738226730346887, 9.865313630026584752182352121278, 10.47495848423784583079309396665
