| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.000897050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.000897050\) |
| \(L(\frac{3}{2})\) |
|
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
−8.735029740799928189809791928130, −7.977286736967013829872605101417, −7.59510584429522567064072154156, −6.09090420539931137386004115631, −5.39283250603867568225700920681, −4.83807003408199520153200414151, −3.70063220993325066814290768909, −2.97441852000047671179891596844, −1.42996832965711412307643694087, −0.64347142892233855644111219742,
1.94738014767813903280436399856, 2.46164978671375085609255082186, 3.57766255148176313338063004705, 4.15760783873136562491476530245, 5.41055927207208593016511386665, 6.53232905104020662115982802768, 7.10861946835602346112218784475, 7.39916903069364600586714711861, 8.501292061228660905617219973303, 9.420266789727902192560820207522
