| L(s) = 1 | + (0.374 − 1.36i)2-s + (1.03 − 1.25i)3-s + (−1.71 − 1.02i)4-s + (−2.03 + 0.933i)5-s + (−1.32 − 1.88i)6-s + (3.03 + 0.985i)7-s + (−2.03 + 1.96i)8-s + (0.0665 + 0.348i)9-s + (0.512 + 3.12i)10-s + (0.494 + 3.91i)11-s + (−3.06 + 1.09i)12-s + (1.16 + 6.13i)13-s + (2.47 − 3.76i)14-s + (−0.936 + 3.51i)15-s + (1.91 + 3.51i)16-s + (0.0165 − 0.0645i)17-s + ⋯ |
| L(s) = 1 | + (0.264 − 0.964i)2-s + (0.598 − 0.723i)3-s + (−0.859 − 0.510i)4-s + (−0.908 + 0.417i)5-s + (−0.539 − 0.768i)6-s + (1.14 + 0.372i)7-s + (−0.719 + 0.694i)8-s + (0.0221 + 0.116i)9-s + (0.162 + 0.986i)10-s + (0.149 + 1.18i)11-s + (−0.883 + 0.316i)12-s + (0.324 + 1.70i)13-s + (0.662 − 1.00i)14-s + (−0.241 + 0.907i)15-s + (0.479 + 0.877i)16-s + (0.00401 − 0.0156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75651 - 0.268567i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75651 - 0.268567i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.374 + 1.36i)T \) |
| 5 | \( 1 + (2.03 - 0.933i)T \) |
| good | 3 | \( 1 + (-1.03 + 1.25i)T + (-0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (-3.03 - 0.985i)T + (5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.494 - 3.91i)T + (-10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (-1.16 - 6.13i)T + (-12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (-0.0165 + 0.0645i)T + (-14.8 - 8.18i)T^{2} \) |
| 19 | \( 1 + (-0.990 + 0.819i)T + (3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-1.62 - 1.72i)T + (-1.44 + 22.9i)T^{2} \) |
| 29 | \( 1 + (9.62 + 0.605i)T + (28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (0.832 + 0.213i)T + (27.1 + 14.9i)T^{2} \) |
| 37 | \( 1 + (3.43 - 1.88i)T + (19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (8.51 + 7.99i)T + (2.57 + 40.9i)T^{2} \) |
| 43 | \( 1 + (3.73 + 2.71i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (2.15 - 5.44i)T + (-34.2 - 32.1i)T^{2} \) |
| 53 | \( 1 + (2.02 + 3.19i)T + (-22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (-5.44 - 2.56i)T + (37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (-10.4 - 11.1i)T + (-3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.309 - 4.92i)T + (-66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (-14.1 - 5.60i)T + (51.7 + 48.6i)T^{2} \) |
| 73 | \( 1 + (-5.00 + 2.35i)T + (46.5 - 56.2i)T^{2} \) |
| 79 | \( 1 + (-6.64 + 8.03i)T + (-14.8 - 77.6i)T^{2} \) |
| 83 | \( 1 + (9.27 + 11.2i)T + (-15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (2.42 + 5.14i)T + (-56.7 + 68.5i)T^{2} \) |
| 97 | \( 1 + (-13.3 - 0.842i)T + (96.2 + 12.1i)T^{2} \) |
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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
−10.01969918218834919305944809930, −8.971935086274592730801059935353, −8.443145713926678857039998134029, −7.43606496419715563049613014405, −6.87087686383242061096576819538, −5.23325925302042532027295291191, −4.44786747012139075490261211173, −3.56662146998474967508554955127, −2.12916283871518315638115187001, −1.70465684374024872049091682739,
0.74634738730031863041203802926, 3.46307550961015008985497750714, 3.60854426370646871600345668426, 4.88911444147228906996454876615, 5.41617107060947458500037747497, 6.70457309035153320720541434100, 7.956689361612116065389582107167, 8.125340034233242729414565243614, 8.813343476418781732112527325155, 9.769546793535639235683440814182
