| L(s) = 1 | + (1.54 + 2.56i)3-s + (0.725 − 1.25i)7-s + (−4.20 + 7.95i)9-s + (8.07 + 4.66i)11-s + (5.60 + 9.70i)13-s − 4.16i·17-s + 3.14·19-s + (4.34 − 0.0803i)21-s + (−11.5 + 6.69i)23-s + (−26.9 + 1.49i)27-s + (12.6 + 7.33i)29-s + (7.07 + 12.2i)31-s + (0.516 + 27.9i)33-s − 18.0·37-s + (−16.2 + 29.4i)39-s + ⋯ |
| L(s) = 1 | + (0.515 + 0.856i)3-s + (0.103 − 0.179i)7-s + (−0.467 + 0.883i)9-s + (0.733 + 0.423i)11-s + (0.430 + 0.746i)13-s − 0.244i·17-s + 0.165·19-s + (0.207 − 0.00382i)21-s + (−0.504 + 0.291i)23-s + (−0.998 + 0.0553i)27-s + (0.437 + 0.252i)29-s + (0.228 + 0.395i)31-s + (0.0156 + 0.847i)33-s − 0.487·37-s + (−0.417 + 0.754i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.119293036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.119293036\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.54 - 2.56i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-0.725 + 1.25i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.07 - 4.66i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-5.60 - 9.70i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 4.16iT - 289T^{2} \) |
| 19 | \( 1 - 3.14T + 361T^{2} \) |
| 23 | \( 1 + (11.5 - 6.69i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-12.6 - 7.33i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-7.07 - 12.2i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 18.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-17.6 + 10.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (16.7 - 28.9i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-1.63 - 0.946i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 97.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (28.9 - 16.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-29.6 + 51.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-47.9 - 83.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 97.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 90.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (66.7 - 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-6.48 - 3.74i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 100. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (32.4 - 56.1i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.06050042210822665663531435357, −9.311216313223253608507601115384, −8.720939354848170141956199607615, −7.74825848365550905285373499444, −6.81308223355572656374718372866, −5.74012639150728372913604130745, −4.59373180984374361651047614290, −3.98414530590085661012218439218, −2.87359911755064355816974652985, −1.53992697888351586761264516686,
0.66376426777999778241456370491, 1.89503518154929343420038473108, 3.08705681241917347460291246515, 4.00015361637800291287252319382, 5.49295063562724655505880385604, 6.29500223752515039324330357111, 7.08440096012848203762282022513, 8.166331613585373543212610128943, 8.543302075514005716248623584995, 9.516490443792210204426894359631
