| L(s) = 1 | + (−1.69 − 0.366i)3-s + 3.38i·5-s + (2.73 + 1.23i)9-s + 2.47·11-s + 6.19·13-s + (1.23 − 5.73i)15-s + 2.47i·17-s − 0.732i·19-s + 6.77·23-s − 6.46·25-s + (−4.17 − 3.09i)27-s + 2.47i·29-s − 9.46i·31-s + (−4.19 − 0.907i)33-s + 4.53·37-s + ⋯ |
| L(s) = 1 | + (−0.977 − 0.211i)3-s + 1.51i·5-s + (0.910 + 0.413i)9-s + 0.747·11-s + 1.71·13-s + (0.319 − 1.48i)15-s + 0.601i·17-s − 0.167i·19-s + 1.41·23-s − 1.29·25-s + (−0.802 − 0.596i)27-s + 0.460i·29-s − 1.69i·31-s + (−0.730 − 0.157i)33-s + 0.745·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.571882958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.571882958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.69 + 0.366i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.38iT - 5T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 6.19T + 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 + 0.732iT - 19T^{2} \) |
| 23 | \( 1 - 6.77T + 23T^{2} \) |
| 29 | \( 1 - 2.47iT - 29T^{2} \) |
| 31 | \( 1 + 9.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 + 9.25iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9.25iT - 53T^{2} \) |
| 59 | \( 1 - 8.34T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 9.25T + 71T^{2} \) |
| 73 | \( 1 - 4.53T + 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 - 8.34T + 83T^{2} \) |
| 89 | \( 1 - 14.2iT - 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−9.163960621506868041299813836941, −8.197572126399322232826416262209, −7.24627569279867929305431353153, −6.63672769987233239184035974705, −6.18422677255951307012815431318, −5.40594420099593269806360705937, −4.07855034348326555080278925616, −3.50611775186291690977950628977, −2.22679910041968144053528955778, −1.00092052762779971811407290502,
0.916627583659023763154438503205, 1.37713764188921359074614637496, 3.35029703891683205445480135684, 4.31867898481347281627546136496, 4.87755097879819219192469035119, 5.69791534468709953312913838367, 6.36556250213853178486191311322, 7.21319702239979826705201304345, 8.363423779998706478831599536910, 8.933664564471651324015149582626
