L(s) = 1 | + (4.70 − 2.71i)2-s + (−0.837 − 1.45i)3-s + (10.7 − 18.5i)4-s + 7.70i·5-s + (−7.87 − 4.54i)6-s + (−13.0 − 7.51i)7-s − 73.1i·8-s + (12.0 − 20.9i)9-s + (20.9 + 36.2i)10-s + (2.17 − 1.25i)11-s − 35.9·12-s − 81.5·14-s + (11.1 − 6.45i)15-s + (−112. − 194. i)16-s + (−1.03 + 1.78i)17-s − 131. i·18-s + ⋯ |
L(s) = 1 | + (1.66 − 0.959i)2-s + (−0.161 − 0.279i)3-s + (1.34 − 2.32i)4-s + 0.689i·5-s + (−0.535 − 0.309i)6-s + (−0.702 − 0.405i)7-s − 3.23i·8-s + (0.448 − 0.776i)9-s + (0.661 + 1.14i)10-s + (0.0595 − 0.0344i)11-s − 0.865·12-s − 1.55·14-s + (0.192 − 0.111i)15-s + (−1.75 − 3.04i)16-s + (−0.0147 + 0.0255i)17-s − 1.71i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.87389 - 3.44282i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87389 - 3.44282i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (-4.70 + 2.71i)T + (4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (0.837 + 1.45i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 - 7.70iT - 125T^{2} \) |
| 7 | \( 1 + (13.0 + 7.51i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-2.17 + 1.25i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (1.03 - 1.78i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-81.7 - 47.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-17.9 - 31.0i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-70.0 - 121. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 264. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (222. - 128. i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-341. + 197. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-128. + 222. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 415. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 504.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-104. - 60.1i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-376. + 651. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-183. + 105. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-355. - 205. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 17.4iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 174.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 963. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (413. - 238. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (906. + 523. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.37422263839907104378371274920, −11.20025479321438675484723323065, −10.40035146141628401697988600135, −9.521028954342913816032151703392, −7.04767560277173963115490359313, −6.48576367953201415051373948764, −5.24878199553386690227466276085, −3.76750089951273494318081626355, −3.03211721797354545617870189231, −1.23700543535223819638544282773,
2.68705197793387958838590065206, 4.16295105478366612052078413812, 5.05197369986610444378690104014, 5.95825632281372819863485309209, 7.12775832859084613411419522576, 8.141903353631335339781854415797, 9.492235534789599649993515216632, 11.12894575132034979887503521685, 12.11816927241830408943813986579, 12.98156548814193165864364654004