| L(s) = 1 | + (−0.755 − 0.654i)2-s + (−0.906 + 1.47i)3-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (1.65 − 0.521i)6-s + (−0.302 − 1.03i)7-s + (0.540 − 0.841i)8-s + (−1.35 − 2.67i)9-s + (0.281 − 0.959i)10-s + (−1.53 − 1.76i)11-s + (−1.58 − 0.687i)12-s + (−0.0134 − 0.00394i)13-s + (−0.446 + 0.976i)14-s + (−1.71 − 0.211i)15-s + (−0.959 + 0.281i)16-s + (0.929 − 6.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.534 − 0.463i)2-s + (−0.523 + 0.852i)3-s + (0.0711 + 0.494i)4-s + (0.185 + 0.406i)5-s + (0.674 − 0.212i)6-s + (−0.114 − 0.389i)7-s + (0.191 − 0.297i)8-s + (−0.452 − 0.892i)9-s + (0.0890 − 0.303i)10-s + (−0.461 − 0.532i)11-s + (−0.458 − 0.198i)12-s + (−0.00372 − 0.00109i)13-s + (−0.119 + 0.261i)14-s + (−0.443 − 0.0546i)15-s + (−0.239 + 0.0704i)16-s + (0.225 − 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.718405 - 0.351302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.718405 - 0.351302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.755 + 0.654i)T \) |
| 3 | \( 1 + (0.906 - 1.47i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (4.70 + 0.910i)T \) |
| good | 7 | \( 1 + (0.302 + 1.03i)T + (-5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (1.53 + 1.76i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.0134 + 0.00394i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.929 + 6.46i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-4.39 + 0.632i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (1.21 + 0.174i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.53 - 4.84i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (5.17 + 2.36i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-10.5 + 4.81i)T + (26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-4.90 - 7.62i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 8.17iT - 47T^{2} \) |
| 53 | \( 1 + (2.01 - 0.590i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-0.995 + 3.39i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-5.07 + 7.88i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.926 + 0.802i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (5.50 + 4.76i)T + (10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (2.10 + 14.6i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-0.0735 + 0.250i)T + (-66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (4.63 - 10.1i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (1.80 - 1.15i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (6.66 - 3.04i)T + (63.5 - 73.3i)T^{2} \) |
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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
−10.29775103736695378507977563329, −9.709103572516429648553028430212, −8.948005416367262840419977223932, −7.79499146670675786026326008484, −6.88091657340908182042304306665, −5.76806880163159891671006655395, −4.80653715940795663081497618720, −3.59029331631764704886285583651, −2.70613521446561894053713481007, −0.60868504136567952221346839390,
1.24757558129876712610473815104, 2.44530400106339843160208383855, 4.37921773320456421836178013345, 5.71955879539679056877508611295, 5.96052155186377838549303842163, 7.22396186198219651768541801351, 7.922797199776286650005982372748, 8.632631806199467467291375519379, 9.771348327904486763601830748274, 10.42845760696925277053570639160
