| L(s) = 1 | + (0.939 + 0.342i)2-s + (1.26 + 1.18i)3-s + (0.766 + 0.642i)4-s + (−1.86 − 2.21i)5-s + (0.779 + 1.54i)6-s + (0.562 − 0.973i)7-s + (0.500 + 0.866i)8-s + (0.184 + 2.99i)9-s + (−0.990 − 2.72i)10-s + (−2.70 + 1.56i)11-s + (0.203 + 1.71i)12-s + (−5.18 − 0.914i)13-s + (0.861 − 0.722i)14-s + (0.283 − 5.01i)15-s + (0.173 + 0.984i)16-s + (0.880 − 2.41i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.728 + 0.685i)3-s + (0.383 + 0.321i)4-s + (−0.832 − 0.992i)5-s + (0.318 + 0.631i)6-s + (0.212 − 0.367i)7-s + (0.176 + 0.306i)8-s + (0.0614 + 0.998i)9-s + (−0.313 − 0.860i)10-s + (−0.816 + 0.471i)11-s + (0.0588 + 0.496i)12-s + (−1.43 − 0.253i)13-s + (0.230 − 0.193i)14-s + (0.0731 − 1.29i)15-s + (0.0434 + 0.246i)16-s + (0.213 − 0.586i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49937 + 0.422423i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49937 + 0.422423i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (-1.26 - 1.18i)T \) |
| 19 | \( 1 + (-4.13 - 1.37i)T \) |
| good | 5 | \( 1 + (1.86 + 2.21i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.562 + 0.973i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.70 - 1.56i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.18 + 0.914i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.880 + 2.41i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.31 + 5.14i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.09 + 0.399i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.90 + 2.25i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 12.0iT - 37T^{2} \) |
| 41 | \( 1 + (-1.06 - 6.02i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.21 + 1.85i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (0.377 + 1.03i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-5.66 - 4.75i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (6.41 + 2.33i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (5.58 + 4.68i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (2.42 + 6.64i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (3.31 - 2.77i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.30 + 7.41i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-4.30 + 0.759i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-12.5 - 7.22i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.38 + 13.5i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (1.47 - 4.04i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−13.75846453081384701797213992170, −12.73551792262923741130553809195, −11.89291890628580048810331166263, −10.50177851568076508910687438052, −9.369925541179830929114753678714, −8.024833909264870499831168241116, −7.43741444401760267841097736770, −5.01397500355329962870533518418, −4.57637462203554667867049529431, −2.92918833894696729200818734994,
2.51966503937493159395211261028, 3.57108468686376618423034821157, 5.45482936014313682332027293784, 7.15508180426884196857886015328, 7.61417069080938147889822805626, 9.184858741823135933705169121097, 10.63653257133874261862829851941, 11.67293472169130634079366014014, 12.47508636012342895280108532769, 13.55325290301485014343464609412
