| L(s) = 1 | + (−0.841 − 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (−0.415 + 0.909i)6-s + (1.32 − 1.53i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)10-s + (−4.19 + 2.69i)11-s + (0.841 − 0.540i)12-s + (−3.70 − 4.27i)13-s + (−1.94 + 0.570i)14-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (1.34 − 2.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (0.429 + 0.125i)5-s + (−0.169 + 0.371i)6-s + (0.501 − 0.578i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.207 − 0.238i)10-s + (−1.26 + 0.813i)11-s + (0.242 − 0.156i)12-s + (−1.02 − 1.18i)13-s + (−0.519 + 0.152i)14-s + (0.0367 − 0.255i)15-s + (−0.163 + 0.188i)16-s + (0.327 − 0.716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.222251 - 0.769502i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.222251 - 0.769502i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.78 - 0.236i)T \) |
| good | 7 | \( 1 + (-1.32 + 1.53i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.19 - 2.69i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.70 + 4.27i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.34 + 2.95i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.34 + 2.93i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.365 + 0.799i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.55 + 10.7i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (6.98 - 2.05i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (6.34 + 1.86i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (1.16 + 8.11i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 3.29T + 47T^{2} \) |
| 53 | \( 1 + (2.45 - 2.83i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.20 - 4.85i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.21 + 8.45i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (8.99 + 5.77i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (9.47 + 6.08i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.92 - 15.1i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (1.28 + 1.48i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (5.00 - 1.46i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.22 - 15.4i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-17.2 - 5.06i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.24808049262279526847398707769, −9.420584922396485802422052763334, −8.230357109426186659895152206315, −7.47980619833080225620672686750, −7.03838428907927292520830115358, −5.50182082150404992702262959763, −4.73594508578360088053219118018, −2.94900003273198676372326749690, −2.13552083657966098342465479384, −0.49424694794859905229440066781,
1.79569634351265824979081067276, 3.08882171029414034976129920791, 4.80441195655582782693131118191, 5.35142066932625289406507405147, 6.35530828106242230561887702142, 7.43533797013974994924375887659, 8.576019504075858404788681862679, 8.841557032051556293610924024449, 10.09125210673106291822394604333, 10.46944991692167418128681721101
