| L(s) = 1 | − 3-s − 4.29i·5-s + 9-s + 5.06i·11-s − 3.37i·13-s + 4.29i·15-s − 2.76i·17-s − 4.70·19-s − 4.83i·23-s − 13.4·25-s − 27-s + 2.46·29-s + 5.69·31-s − 5.06i·33-s − 2.33·37-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.92i·5-s + 0.333·9-s + 1.52i·11-s − 0.937i·13-s + 1.10i·15-s − 0.670i·17-s − 1.07·19-s − 1.00i·23-s − 2.69·25-s − 0.192·27-s + 0.456·29-s + 1.02·31-s − 0.881i·33-s − 0.383·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5053848647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5053848647\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.29iT - 5T^{2} \) |
| 11 | \( 1 - 5.06iT - 11T^{2} \) |
| 13 | \( 1 + 3.37iT - 13T^{2} \) |
| 17 | \( 1 + 2.76iT - 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 23 | \( 1 + 4.83iT - 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 - 5.69T + 31T^{2} \) |
| 37 | \( 1 + 2.33T + 37T^{2} \) |
| 41 | \( 1 - 5.14iT - 41T^{2} \) |
| 43 | \( 1 + 13.0iT - 43T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 + 4.22T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 - 3.87iT - 61T^{2} \) |
| 67 | \( 1 + 4.76iT - 67T^{2} \) |
| 71 | \( 1 - 12.3iT - 71T^{2} \) |
| 73 | \( 1 + 11.5iT - 73T^{2} \) |
| 79 | \( 1 - 13.9iT - 79T^{2} \) |
| 83 | \( 1 + 7.32T + 83T^{2} \) |
| 89 | \( 1 - 14.0iT - 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−8.496961800203769550342301158909, −7.977000521493754498127070233816, −6.98758961684110350320043857798, −6.10190163031129896291692729212, −5.08396372621796667810403286592, −4.77830647290975581400262329014, −4.05440392729150360392746425242, −2.38368569404373085683774759423, −1.28948647527136326542998099411, −0.19063423528115897282469064152,
1.72196845942609041787124138878, 2.91377190868148287189639842626, 3.57708235973686448729513228312, 4.54959383177740072759358262544, 6.01924517448113475142263181830, 6.17558615101279819090035246479, 6.89087627903718080047936157187, 7.76549994429278981884314836600, 8.547353287816585143836000723437, 9.602175010997593231847971082481
