| L(s) = 1 | + (0.159 − 1.40i)2-s + (−3.21 + 0.281i)3-s + (−1.94 − 0.449i)4-s + (−2.71 − 1.26i)5-s + (−0.118 + 4.56i)6-s + (−0.223 + 0.129i)7-s + (−0.942 + 2.66i)8-s + (7.31 − 1.28i)9-s + (−2.21 + 3.61i)10-s + (1.33 − 0.356i)11-s + (6.39 + 0.895i)12-s + (−0.197 + 2.25i)13-s + (0.145 + 0.335i)14-s + (9.09 + 3.31i)15-s + (3.59 + 1.75i)16-s + (1.28 − 7.26i)17-s + ⋯ |
| L(s) = 1 | + (0.112 − 0.993i)2-s + (−1.85 + 0.162i)3-s + (−0.974 − 0.224i)4-s + (−1.21 − 0.566i)5-s + (−0.0483 + 1.86i)6-s + (−0.0845 + 0.0488i)7-s + (−0.333 + 0.942i)8-s + (2.43 − 0.429i)9-s + (−0.700 + 1.14i)10-s + (0.401 − 0.107i)11-s + (1.84 + 0.258i)12-s + (−0.0547 + 0.625i)13-s + (0.0389 + 0.0895i)14-s + (2.34 + 0.854i)15-s + (0.899 + 0.437i)16-s + (0.310 − 1.76i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.297398 + 0.0475700i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.297398 + 0.0475700i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.159 + 1.40i)T \) |
| 19 | \( 1 + (2.33 - 3.67i)T \) |
| good | 3 | \( 1 + (3.21 - 0.281i)T + (2.95 - 0.520i)T^{2} \) |
| 5 | \( 1 + (2.71 + 1.26i)T + (3.21 + 3.83i)T^{2} \) |
| 7 | \( 1 + (0.223 - 0.129i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.33 + 0.356i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (0.197 - 2.25i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.28 + 7.26i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (2.75 - 7.55i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-4.36 - 3.05i)T + (9.91 + 27.2i)T^{2} \) |
| 31 | \( 1 + (-1.07 - 1.86i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.78 - 3.78i)T + 37iT^{2} \) |
| 41 | \( 1 + (2.55 - 3.04i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.72 + 1.27i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (0.508 + 2.88i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (0.525 + 1.12i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-6.10 + 4.27i)T + (20.1 - 55.4i)T^{2} \) |
| 61 | \( 1 + (-4.06 + 1.89i)T + (39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (4.64 + 3.25i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-2.76 - 7.59i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (8.36 - 9.97i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-11.7 - 9.88i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.91 - 0.781i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.609 + 0.726i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (1.81 - 10.3i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.75618378008315470251240123541, −11.31605664084376084888961300508, −10.14937160635485844448221979056, −9.375161657747703452615895086675, −7.970808091211758528733636419994, −6.69966808877775833450705737058, −5.38115105437838317723095949514, −4.63570886852145237354260379242, −3.71959629713437775202332285567, −1.11677781540469980560860815195,
0.34156802738438913087008236482, 3.93322706155648474844750179616, 4.69137850725800989126975877533, 6.05341580469524240803428901987, 6.54235803780276718078546109199, 7.51769186241696523425713322381, 8.409007440901305803935157406452, 10.16696530345761372971560073810, 10.74427206459106879181619413997, 11.82085045847867105143033918232
