| L(s) = 1 | + (0.496 + 0.226i)3-s + (−1.16 − 1.00i)5-s + (−0.981 + 0.630i)7-s + (−1.76 − 2.04i)9-s + (4.28 + 0.616i)11-s + (1.76 − 2.74i)13-s + (−0.349 − 0.765i)15-s + (−3.79 − 1.11i)17-s + (0.469 + 1.59i)19-s + (−0.630 + 0.0906i)21-s + (1.22 − 4.63i)23-s + (−0.373 − 2.59i)25-s + (−0.877 − 2.98i)27-s + (1.96 − 6.68i)29-s + (−1.28 − 2.81i)31-s + ⋯ |
| L(s) = 1 | + (0.286 + 0.131i)3-s + (−0.520 − 0.451i)5-s + (−0.370 + 0.238i)7-s + (−0.589 − 0.680i)9-s + (1.29 + 0.185i)11-s + (0.489 − 0.761i)13-s + (−0.0903 − 0.197i)15-s + (−0.919 − 0.270i)17-s + (0.107 + 0.366i)19-s + (−0.137 + 0.0197i)21-s + (0.254 − 0.966i)23-s + (−0.0746 − 0.519i)25-s + (−0.168 − 0.575i)27-s + (0.364 − 1.24i)29-s + (−0.230 − 0.505i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.990206 - 0.802286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.990206 - 0.802286i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-1.22 + 4.63i)T \) |
| good | 3 | \( 1 + (-0.496 - 0.226i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (1.16 + 1.00i)T + (0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.981 - 0.630i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-4.28 - 0.616i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.76 + 2.74i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (3.79 + 1.11i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.469 - 1.59i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.96 + 6.68i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.28 + 2.81i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.91 + 2.52i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.80 + 5.54i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.605 + 0.276i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 + (-5.72 - 8.90i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.863 + 1.34i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.81 + 0.829i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (2.22 - 0.320i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.69 - 11.7i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (6.66 - 1.95i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-4.03 - 2.59i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-13.6 + 11.8i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.74 + 16.9i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (0.978 - 1.12i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.04803249028155922603405703418, −9.127770800184212829575738746402, −8.681178321863716150181924989739, −7.76521489359476150053814457693, −6.50337117668044761044670126871, −5.94346990001622848030899146182, −4.45346409279402953003721469968, −3.75208183994385153909644916484, −2.55150564339516481900880532589, −0.66013800343579508046816486817,
1.62566489958324787841415824390, 3.13093770173824712323988768091, 3.90541200544062273660104752431, 5.10519142967552116380433487185, 6.47678254872029244697840054988, 6.93672010993693598270283463615, 8.023865417992311778982914495323, 8.894765011027684560187386699612, 9.498828166805979496451664382004, 10.85436188891566576274937566092
