| L(s) = 1 | + (−0.573 − 1.29i)2-s + (1.32 − 1.11i)3-s + (−1.34 + 1.48i)4-s + (0.561 − 0.669i)5-s + (−2.20 − 1.07i)6-s + (3.98 + 1.44i)7-s + (2.68 + 0.882i)8-s + (0.520 − 2.95i)9-s + (−1.18 − 0.341i)10-s + (−2.44 − 2.91i)11-s + (−0.128 + 3.46i)12-s + (−0.577 − 0.101i)13-s + (−0.411 − 5.97i)14-s + (−4.05e−5 − 1.51i)15-s + (−0.400 − 3.97i)16-s + (−1.31 + 2.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.405 − 0.913i)2-s + (0.766 − 0.642i)3-s + (−0.670 + 0.741i)4-s + (0.251 − 0.299i)5-s + (−0.898 − 0.439i)6-s + (1.50 + 0.547i)7-s + (0.950 + 0.312i)8-s + (0.173 − 0.984i)9-s + (−0.375 − 0.108i)10-s + (−0.738 − 0.879i)11-s + (−0.0370 + 0.999i)12-s + (−0.160 − 0.0282i)13-s + (−0.109 − 1.59i)14-s + (−1.04e−5 − 0.390i)15-s + (−0.100 − 0.994i)16-s + (−0.319 + 0.554i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0845 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0845 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.896420 - 0.975715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.896420 - 0.975715i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.573 + 1.29i)T \) |
| 3 | \( 1 + (-1.32 + 1.11i)T \) |
| good | 5 | \( 1 + (-0.561 + 0.669i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.98 - 1.44i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (2.44 + 2.91i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.577 + 0.101i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.31 - 2.28i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.73 - 2.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.54 - 0.562i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-6.06 + 1.06i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.38 - 0.503i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.51 - 2.60i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.20 + 6.84i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.80 + 5.72i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.61 - 0.588i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 12.3iT - 53T^{2} \) |
| 59 | \( 1 + (1.82 - 2.17i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.99 - 8.21i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-10.2 - 1.80i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (8.21 - 14.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.34 - 7.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.60 + 14.7i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (14.0 - 2.48i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (4.44 + 7.69i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.68 + 8.13i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.08379691174852655650960400893, −11.15705347526309343638896327823, −10.19777697767008866464758655212, −8.665265752888543769434673419549, −8.538190897792954204266736737742, −7.53220691551617193432845363059, −5.67134273314046513738879437045, −4.24573920629103286539715570490, −2.63648432082727454800895784196, −1.54140480507137660836828469020,
2.17445437327796657953658156493, 4.50014810878771597337763587341, 4.91270488082964757586201065247, 6.71984576353503013035424395559, 7.83688005100351512383905015057, 8.351249494604134121704146353297, 9.578436598169162570491135074864, 10.40430006505687770015691764078, 11.12679234918490616302922361500, 12.97759166976426666623926761936
