L(s) = 1 | + (1.62 − 0.586i)3-s + (−0.347 − 1.96i)4-s + (−0.764 + 2.10i)5-s + (−0.459 + 2.60i)7-s + (2.31 − 1.91i)9-s + (2.20 + 6.07i)11-s + (−1.72 − 3.00i)12-s + (−3.66 + 3.07i)13-s + (−0.0137 + 3.87i)15-s + (−3.75 + 1.36i)16-s + (−6.43 + 3.71i)17-s + (4.40 + 0.776i)20-s + (0.779 + 4.51i)21-s + (−3.83 − 3.21i)25-s + (2.64 − 4.47i)27-s + 5.29·28-s + ⋯ |
L(s) = 1 | + (0.940 − 0.338i)3-s + (−0.173 − 0.984i)4-s + (−0.342 + 0.939i)5-s + (−0.173 + 0.984i)7-s + (0.770 − 0.637i)9-s + (0.666 + 1.83i)11-s + (−0.496 − 0.867i)12-s + (−1.01 + 0.852i)13-s + (−0.00354 + 0.999i)15-s + (−0.939 + 0.342i)16-s + (−1.56 + 0.900i)17-s + (0.984 + 0.173i)20-s + (0.170 + 0.985i)21-s + (−0.766 − 0.642i)25-s + (0.509 − 0.860i)27-s + 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22025 + 0.954356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22025 + 0.954356i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.62 + 0.586i)T \) |
| 5 | \( 1 + (0.764 - 2.10i)T \) |
| 7 | \( 1 + (0.459 - 2.60i)T \) |
good | 2 | \( 1 + (0.347 + 1.96i)T^{2} \) |
| 11 | \( 1 + (-2.20 - 6.07i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.66 - 3.07i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (6.43 - 3.71i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.10 + 3.70i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-13.0 - 2.29i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-12.0 + 6.93i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.68 + 9.85i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.15 - 6.84i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.15 + 2.56i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.22 - 3.35i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.935474536962757713009252948259, −9.439906989767752764434538804454, −8.786559651436862044426042868623, −7.57256564143212175824365072971, −6.71269237030343510271876529939, −6.35557003815050612547419524922, −4.71550024212255112044410233917, −4.04513812182541576505533194066, −2.27174985008607361999612047431, −2.09155794312357655235887297366,
0.62599268018819144891395713597, 2.66346118819429920313660608297, 3.58206084741453424397585170447, 4.26761044509113995917967072238, 5.14168611434839543268547262391, 6.80628020458045071978220156687, 7.57199476160779041525216261353, 8.357103464645626055140458369548, 8.866897393226400721294284317389, 9.541087822528184304430898445269