L(s) = 1 | + (0.124 + 0.464i)2-s + (−0.965 − 0.258i)3-s + (1.53 − 0.884i)4-s − 0.480i·6-s + (−1.13 + 2.38i)7-s + (1.28 + 1.28i)8-s + (0.866 + 0.499i)9-s + (1.38 + 2.39i)11-s + (−1.70 + 0.457i)12-s + (0.707 − 0.707i)13-s + (−1.25 − 0.231i)14-s + (1.33 − 2.30i)16-s + (−1.14 + 4.28i)17-s + (−0.124 + 0.464i)18-s + (−0.280 + 0.486i)19-s + ⋯ |
L(s) = 1 | + (0.0880 + 0.328i)2-s + (−0.557 − 0.149i)3-s + (0.765 − 0.442i)4-s − 0.196i·6-s + (−0.430 + 0.902i)7-s + (0.453 + 0.453i)8-s + (0.288 + 0.166i)9-s + (0.417 + 0.722i)11-s + (−0.493 + 0.132i)12-s + (0.196 − 0.196i)13-s + (−0.334 − 0.0618i)14-s + (0.333 − 0.577i)16-s + (−0.278 + 1.03i)17-s + (−0.0293 + 0.109i)18-s + (−0.0643 + 0.111i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39470 + 0.596564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39470 + 0.596564i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.13 - 2.38i)T \) |
good | 2 | \( 1 + (-0.124 - 0.464i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (-1.38 - 2.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 0.707i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.14 - 4.28i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.280 - 0.486i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.77 + 2.08i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 5.53iT - 29T^{2} \) |
| 31 | \( 1 + (-6.97 + 4.02i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.09 + 4.09i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 4.21iT - 41T^{2} \) |
| 43 | \( 1 + (-1.08 - 1.08i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.46 - 2.26i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.194 + 0.727i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-5.09 - 8.81i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.65 - 2.58i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + (7.60 + 2.03i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-8.05 - 4.65i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.20 + 6.20i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.67 + 13.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.41 + 6.41i)T + 97iT^{2} \) |
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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
−10.99582204195181801320390133365, −10.24303297779744206845543292020, −9.254818973986090943708083031647, −8.199047494274310535135524832286, −7.00859433185883322935625770447, −6.39967974934758827696273034343, −5.61362730014660165743152943252, −4.59887354134091536702220788418, −2.87661225136349458431603224017, −1.56629606871687372685580038538,
1.04241162680889422622471840881, 2.89008526245694080810980298844, 3.82800569077864091711754068909, 4.97994493584559310792081668766, 6.45458364796594804570890938293, 6.86073441533332534997910297469, 7.899578185155338761882213318708, 9.134743603108980552019499317514, 10.12129620075117170605361509927, 10.93489630970651059901405976098