L(s) = 1 | + (−1.26 − 0.622i)2-s + (0.562 + 0.376i)3-s + (1.22 + 1.58i)4-s + (1.23 + 1.86i)5-s + (−0.480 − 0.828i)6-s + (0.396 − 0.164i)7-s + (−0.568 − 2.77i)8-s + (−0.972 − 2.34i)9-s + (−0.405 − 3.13i)10-s + (3.28 + 0.653i)11-s + (0.0938 + 1.35i)12-s + (0.950 + 4.77i)13-s + (−0.604 − 0.0384i)14-s + (−0.00643 + 1.51i)15-s + (−1.00 + 3.87i)16-s − 3.35·17-s + ⋯ |
L(s) = 1 | + (−0.897 − 0.440i)2-s + (0.324 + 0.217i)3-s + (0.611 + 0.790i)4-s + (0.552 + 0.833i)5-s + (−0.196 − 0.338i)6-s + (0.149 − 0.0620i)7-s + (−0.200 − 0.979i)8-s + (−0.324 − 0.782i)9-s + (−0.128 − 0.991i)10-s + (0.990 + 0.197i)11-s + (0.0271 + 0.389i)12-s + (0.263 + 1.32i)13-s + (−0.161 − 0.0102i)14-s + (−0.00166 + 0.390i)15-s + (−0.251 + 0.967i)16-s − 0.812·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06768 + 0.251683i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06768 + 0.251683i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.26 + 0.622i)T \) |
| 5 | \( 1 + (-1.23 - 1.86i)T \) |
good | 3 | \( 1 + (-0.562 - 0.376i)T + (1.14 + 2.77i)T^{2} \) |
| 7 | \( 1 + (-0.396 + 0.164i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-3.28 - 0.653i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-0.950 - 4.77i)T + (-12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 + (-2.30 - 1.54i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (0.570 - 1.37i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-3.65 + 0.727i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 9.38T + 31T^{2} \) |
| 37 | \( 1 + (4.14 + 0.823i)T + (34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.549 + 1.32i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-1.00 - 1.50i)T + (-16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + 0.982iT - 47T^{2} \) |
| 53 | \( 1 + (3.34 + 5.00i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (4.02 + 6.02i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (0.761 - 0.151i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (2.60 - 3.89i)T + (-25.6 - 61.8i)T^{2} \) |
| 71 | \( 1 + (-5.09 + 12.3i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-5.78 + 13.9i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-7.34 - 7.34i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.49 + 7.52i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (14.2 + 5.92i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (7.61 - 7.61i)T - 97iT^{2} \) |
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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
−11.60836973284876453922389093551, −10.69758061245820238378803678169, −9.568439385456015296525292346710, −9.264909251294231544067494237862, −8.157139696030060683573943576804, −6.73188342487115780810266253409, −6.41153618679087661476299358645, −4.17136706730358100142890481895, −3.08040338547015219572860802075, −1.71387476451845938549599261581,
1.17247908330470269196660323148, 2.63349468810626748568523441247, 4.81387485642035757502558335415, 5.78596679336257025426124543666, 6.81934844387132506836029273958, 8.192530409304294301599892126563, 8.499916720684379648134607480357, 9.515460190316387803824030011057, 10.42974748549880635129578679454, 11.37101339999065018930718829612