| L(s) = 1 | + (−0.851 + 0.523i)2-s + (0.655 − 1.75i)3-s + (0.451 − 0.892i)4-s + (0.237 + 0.322i)5-s + (0.358 + 1.83i)6-s + (−1.22 + 1.87i)7-s + (0.0825 + 0.996i)8-s + (−0.374 − 0.325i)9-s + (−0.370 − 0.150i)10-s + (−3.25 − 1.11i)11-s + (−1.26 − 1.37i)12-s + (−3.48 − 4.23i)13-s + (0.0618 − 2.24i)14-s + (0.720 − 0.203i)15-s + (−0.592 − 0.805i)16-s + (−1.59 − 3.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.602 + 0.370i)2-s + (0.378 − 1.01i)3-s + (0.225 − 0.446i)4-s + (0.106 + 0.144i)5-s + (0.146 + 0.749i)6-s + (−0.464 + 0.710i)7-s + (0.0291 + 0.352i)8-s + (−0.124 − 0.108i)9-s + (−0.117 − 0.0476i)10-s + (−0.981 − 0.336i)11-s + (−0.365 − 0.397i)12-s + (−0.966 − 1.17i)13-s + (0.0165 − 0.599i)14-s + (0.185 − 0.0525i)15-s + (−0.148 − 0.201i)16-s + (−0.386 − 0.764i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.178192 - 0.538393i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.178192 - 0.538393i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.851 - 0.523i)T \) |
| 19 | \( 1 + (3.63 - 2.41i)T \) |
| good | 3 | \( 1 + (-0.655 + 1.75i)T + (-2.26 - 1.97i)T^{2} \) |
| 5 | \( 1 + (-0.237 - 0.322i)T + (-1.49 + 4.77i)T^{2} \) |
| 7 | \( 1 + (1.22 - 1.87i)T + (-2.81 - 6.41i)T^{2} \) |
| 11 | \( 1 + (3.25 + 1.11i)T + (8.68 + 6.75i)T^{2} \) |
| 13 | \( 1 + (3.48 + 4.23i)T + (-2.49 + 12.7i)T^{2} \) |
| 17 | \( 1 + (1.59 + 3.15i)T + (-10.0 + 13.6i)T^{2} \) |
| 23 | \( 1 + (1.85 + 4.94i)T + (-17.3 + 15.1i)T^{2} \) |
| 29 | \( 1 + (-1.69 - 0.0933i)T + (28.8 + 3.19i)T^{2} \) |
| 31 | \( 1 + (0.225 - 0.122i)T + (16.9 - 25.9i)T^{2} \) |
| 37 | \( 1 + (3.71 + 1.27i)T + (29.1 + 22.7i)T^{2} \) |
| 41 | \( 1 + (5.44 + 5.29i)T + (1.12 + 40.9i)T^{2} \) |
| 43 | \( 1 + (1.01 - 0.414i)T + (30.8 - 29.9i)T^{2} \) |
| 47 | \( 1 + (-0.434 - 2.22i)T + (-43.5 + 17.6i)T^{2} \) |
| 53 | \( 1 + (1.19 + 6.09i)T + (-49.1 + 19.9i)T^{2} \) |
| 59 | \( 1 + (-7.34 - 7.14i)T + (1.62 + 58.9i)T^{2} \) |
| 61 | \( 1 + (8.86 - 4.18i)T + (38.7 - 47.0i)T^{2} \) |
| 67 | \( 1 + (2.66 - 1.84i)T + (23.4 - 62.7i)T^{2} \) |
| 71 | \( 1 + (-2.75 - 1.29i)T + (45.1 + 54.8i)T^{2} \) |
| 73 | \( 1 + (1.03 + 2.05i)T + (-43.2 + 58.8i)T^{2} \) |
| 79 | \( 1 + (1.92 - 0.780i)T + (56.6 - 55.0i)T^{2} \) |
| 83 | \( 1 + (1.52 + 3.46i)T + (-56.2 + 61.0i)T^{2} \) |
| 89 | \( 1 + (-5.46 + 10.8i)T + (-52.7 - 71.7i)T^{2} \) |
| 97 | \( 1 + (-10.0 - 6.96i)T + (34.0 + 90.8i)T^{2} \) |
| 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
−10.30732108865580217216975036772, −8.898632775556189584091070717921, −8.257293095320320758972355899471, −7.56408897783800250996024400894, −6.71833962399167102998343648342, −5.87385476569880325478829652781, −4.85733092587708058605970854047, −2.80991777185758908265067124050, −2.24602002679382861179903073175, −0.31350707823148748598837089176,
1.92363017600700199430772979914, 3.24117936462796529232444718465, 4.20226727479742132669141995927, 4.99892060805008569386663256033, 6.62252404549024318753674784715, 7.34975936011340955824542851664, 8.455428785941315377688095654798, 9.297538938625069320618368227379, 9.877641179839630950587358509473, 10.45451273292429745251099701819
