| L(s) = 1 | + (0.557 + 1.29i)2-s + (0.349 + 1.75i)3-s + (−1.37 + 1.44i)4-s + (1.08 − 1.62i)5-s + (−2.09 + 1.43i)6-s + (0.680 + 1.01i)7-s + (−2.65 − 0.983i)8-s + (−0.199 + 0.0825i)9-s + (2.72 + 0.505i)10-s + (−3.15 − 0.628i)11-s + (−3.03 − 1.91i)12-s + (−0.472 − 0.472i)13-s + (−0.944 + 1.45i)14-s + (3.24 + 1.34i)15-s + (−0.201 − 3.99i)16-s + (0.972 − 4.00i)17-s + ⋯ |
| L(s) = 1 | + (0.394 + 0.918i)2-s + (0.201 + 1.01i)3-s + (−0.689 + 0.724i)4-s + (0.486 − 0.727i)5-s + (−0.853 + 0.585i)6-s + (0.257 + 0.385i)7-s + (−0.937 − 0.347i)8-s + (−0.0664 + 0.0275i)9-s + (0.860 + 0.159i)10-s + (−0.952 − 0.189i)11-s + (−0.875 − 0.553i)12-s + (−0.131 − 0.131i)13-s + (−0.252 + 0.388i)14-s + (0.837 + 0.346i)15-s + (−0.0502 − 0.998i)16-s + (0.235 − 0.971i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.793556 + 1.11304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.793556 + 1.11304i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.557 - 1.29i)T \) |
| 17 | \( 1 + (-0.972 + 4.00i)T \) |
| good | 3 | \( 1 + (-0.349 - 1.75i)T + (-2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-1.08 + 1.62i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.680 - 1.01i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (3.15 + 0.628i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (0.472 + 0.472i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.347 + 0.143i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-7.81 - 1.55i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (0.263 + 0.176i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (0.751 + 3.77i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (6.83 - 1.35i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (5.01 - 3.35i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (7.39 - 3.06i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (6.28 + 6.28i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.69 + 11.3i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (1.04 + 2.52i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (9.24 - 6.17i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 - 1.57iT - 67T^{2} \) |
| 71 | \( 1 + (9.54 - 1.89i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-3.44 - 2.30i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (2.67 - 13.4i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-2.64 + 6.38i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-11.0 - 11.0i)T + 89iT^{2} \) |
| 97 | \( 1 + (2.39 - 3.58i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−13.49698412095565758052480312964, −12.94621038295541705007388988394, −11.61493911820630035008793705588, −10.09093904992149204929415467475, −9.210710545160189369258403280056, −8.401148626568811429075070892392, −7.02953007404619136488320335421, −5.24648180886052415418329490101, −4.97993329608399438561061673797, −3.25926483754614221178567648468,
1.71887003285825118646649757416, 3.02165381679500816722781783379, 4.85738271388986764114851832194, 6.33226582850145874721944137501, 7.44899558499707720657716619360, 8.764834627733166396730780226620, 10.36070776024756514706690664594, 10.67490126649015632223853503486, 12.15668648175538238918824508279, 12.93492153868922153859594533176
