| L(s) = 1 | + (−1.28 − 0.589i)2-s + (−0.760 − 1.55i)3-s + (1.30 + 1.51i)4-s + (−1.70 + 1.44i)5-s + (0.0608 + 2.44i)6-s + 1.06·7-s + (−0.787 − 2.71i)8-s + (−1.84 + 2.36i)9-s + (3.04 − 0.853i)10-s + (3.04 + 2.21i)11-s + (1.36 − 3.18i)12-s + (0.659 + 0.908i)13-s + (−1.36 − 0.627i)14-s + (3.54 + 1.55i)15-s + (−0.588 + 3.95i)16-s + (−0.888 + 2.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.909 − 0.416i)2-s + (−0.438 − 0.898i)3-s + (0.653 + 0.757i)4-s + (−0.762 + 0.646i)5-s + (0.0248 + 0.999i)6-s + 0.402·7-s + (−0.278 − 0.960i)8-s + (−0.614 + 0.788i)9-s + (0.962 − 0.269i)10-s + (0.917 + 0.666i)11-s + (0.393 − 0.919i)12-s + (0.183 + 0.251i)13-s + (−0.366 − 0.167i)14-s + (0.915 + 0.401i)15-s + (−0.147 + 0.989i)16-s + (−0.215 + 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.680270 - 0.0228231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.680270 - 0.0228231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 + 0.589i)T \) |
| 3 | \( 1 + (0.760 + 1.55i)T \) |
| 5 | \( 1 + (1.70 - 1.44i)T \) |
| good | 7 | \( 1 - 1.06T + 7T^{2} \) |
| 11 | \( 1 + (-3.04 - 2.21i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.659 - 0.908i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.888 - 2.73i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-4.04 - 1.31i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.00 + 5.50i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.10 + 0.685i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.361 + 0.117i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.98 - 9.61i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.81 - 3.87i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 9.24T + 43T^{2} \) |
| 47 | \( 1 + (-7.50 + 2.44i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.84 - 5.67i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (5.06 - 3.67i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.73 - 1.98i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (2.81 - 8.65i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.55 - 4.79i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.951 - 1.30i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-13.0 + 4.22i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (14.7 + 4.80i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-4.85 + 6.68i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (11.0 - 3.58i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.69135506150866932617021537603, −10.98411624972440553823208166136, −10.03407764209473444292428378030, −8.686340548759898537118385420663, −7.903244596337949044657460838502, −7.00374895592265167336786481962, −6.35184857826070618479679160114, −4.36944975690319293855057266682, −2.82955785110858182724040150605, −1.31805527633011373100890027146,
0.856662238765732596709290245371, 3.43485894827721837892909191439, 4.85588357537948332426063996870, 5.71231485901471918691904595610, 7.03369160566791336092231991188, 8.110332813063534620184177143153, 9.107710065371073449236706438189, 9.513305970125152140069616365894, 11.04209297719950273875399366582, 11.30883857693017969789879088631
