L(s) = 1 | + (−0.584 − 1.28i)2-s + 2.81i·3-s + (−1.31 + 1.50i)4-s + (0.390 + 2.20i)5-s + (3.62 − 1.64i)6-s + 4.13·7-s + (2.70 + 0.813i)8-s − 4.91·9-s + (2.60 − 1.78i)10-s + 0.681i·11-s + (−4.23 − 3.70i)12-s − 3.19·13-s + (−2.41 − 5.31i)14-s + (−6.19 + 1.09i)15-s + (−0.535 − 3.96i)16-s + 2.93i·17-s + ⋯ |
L(s) = 1 | + (−0.413 − 0.910i)2-s + 1.62i·3-s + (−0.658 + 0.752i)4-s + (0.174 + 0.984i)5-s + (1.47 − 0.671i)6-s + 1.56·7-s + (0.957 + 0.287i)8-s − 1.63·9-s + (0.824 − 0.565i)10-s + 0.205i·11-s + (−1.22 − 1.06i)12-s − 0.886·13-s + (−0.645 − 1.42i)14-s + (−1.59 + 0.283i)15-s + (−0.133 − 0.990i)16-s + 0.711i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00513 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00513 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.801378 + 0.797275i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.801378 + 0.797275i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.584 + 1.28i)T \) |
| 5 | \( 1 + (-0.390 - 2.20i)T \) |
| 19 | \( 1 + (2.23 + 3.74i)T \) |
good | 3 | \( 1 - 2.81iT - 3T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 - 0.681iT - 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 - 2.93iT - 17T^{2} \) |
| 23 | \( 1 - 6.41T + 23T^{2} \) |
| 29 | \( 1 - 0.928iT - 29T^{2} \) |
| 31 | \( 1 - 6.68T + 31T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 + 4.89iT - 41T^{2} \) |
| 43 | \( 1 + 7.02T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 - 7.60T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 - 5.12iT - 67T^{2} \) |
| 71 | \( 1 - 7.43T + 71T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 9.74T + 83T^{2} \) |
| 89 | \( 1 + 7.65iT - 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−11.26080610851005353639980645480, −10.54728349788069897295583766224, −10.13855808948006487587289204400, −9.058131965845935767696068470696, −8.298557377909835388977374837766, −7.09374158893630331456668451678, −5.13864895794901414071853146325, −4.57376180271174859675307448399, −3.40269887062992252750317404148, −2.20627932182737561489901444489,
0.942916199995733321929266484114, 1.96415551418379009857766963662, 4.74927304105948917302431583248, 5.38040964189208813471768784558, 6.59998698516896449459427574687, 7.50111844474357587866076640079, 8.223265205607326618317131814740, 8.686909028605270038654747812636, 9.984106564017579526939146018590, 11.37995024549317219966684803578