| L(s) = 1 | + (−1.72 − 2.37i)3-s + (6.46 + 9.12i)5-s + 7.35i·7-s + (5.68 − 17.4i)9-s + (14.2 + 43.8i)11-s + (51.4 + 16.7i)13-s + (10.5 − 31.0i)15-s + (63.8 − 87.9i)17-s + (112. + 81.8i)19-s + (17.4 − 12.6i)21-s + (−97.8 + 31.7i)23-s + (−41.4 + 117. i)25-s + (−126. + 41.1i)27-s + (72.5 − 52.7i)29-s + (−88.2 − 64.1i)31-s + ⋯ |
| L(s) = 1 | + (−0.331 − 0.456i)3-s + (0.578 + 0.815i)5-s + 0.397i·7-s + (0.210 − 0.647i)9-s + (0.390 + 1.20i)11-s + (1.09 + 0.356i)13-s + (0.180 − 0.534i)15-s + (0.911 − 1.25i)17-s + (1.36 + 0.988i)19-s + (0.181 − 0.131i)21-s + (−0.887 + 0.288i)23-s + (−0.331 + 0.943i)25-s + (−0.902 + 0.293i)27-s + (0.464 − 0.337i)29-s + (−0.511 − 0.371i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.60289 + 0.285945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60289 + 0.285945i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-6.46 - 9.12i)T \) |
| good | 3 | \( 1 + (1.72 + 2.37i)T + (-8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 - 7.35iT - 343T^{2} \) |
| 11 | \( 1 + (-14.2 - 43.8i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-51.4 - 16.7i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-63.8 + 87.9i)T + (-1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-112. - 81.8i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (97.8 - 31.7i)T + (9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-72.5 + 52.7i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (88.2 + 64.1i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (199. + 64.9i)T + (4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (134. - 415. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 289. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (148. + 204. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (197. + 271. i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (80.7 - 248. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (117. + 360. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (190. - 262. i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (572. - 416. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-875. + 284. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-550. + 399. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-542. + 747. i)T + (-1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-196. - 604. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (308. + 423. i)T + (-2.82e5 + 8.68e5i)T^{2} \) |
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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
−13.55258947925486707149337324323, −12.13936979118056568661241320345, −11.65553505334161417358095144106, −10.05010112792040858882026375420, −9.388839041927174342860010266781, −7.56822550663357992030093101833, −6.60421725940232031506550860365, −5.55491906435475002035447064605, −3.51461220381981457513871898030, −1.62241664522098717544120036572,
1.19492429218881007897296651836, 3.66763042401163326380813521739, 5.18239997657702341665805285607, 6.11118226652335661548965824315, 7.964767465724150649215288778608, 8.968103437852266505146167077392, 10.24560714170071643630295620487, 11.00326263557652797747184377614, 12.30770805179607311294080153084, 13.53790137765370875291049367469
