L(s) = 1 | + (−0.939 + 0.342i)3-s + (−0.134 + 0.764i)5-s + (−0.911 + 0.526i)7-s + (0.766 − 0.642i)9-s + (3.03 + 1.75i)11-s + (1.81 − 4.99i)13-s + (−0.134 − 0.764i)15-s + (−0.231 − 0.194i)17-s + (−0.105 + 4.35i)19-s + (0.676 − 0.806i)21-s + (−5.32 + 0.938i)23-s + (4.13 + 1.50i)25-s + (−0.500 + 0.866i)27-s + (1.86 + 2.22i)29-s + (2.52 + 4.38i)31-s + ⋯ |
L(s) = 1 | + (−0.542 + 0.197i)3-s + (−0.0602 + 0.341i)5-s + (−0.344 + 0.198i)7-s + (0.255 − 0.214i)9-s + (0.916 + 0.529i)11-s + (0.504 − 1.38i)13-s + (−0.0347 − 0.197i)15-s + (−0.0561 − 0.0471i)17-s + (−0.0243 + 0.999i)19-s + (0.147 − 0.175i)21-s + (−1.11 + 0.195i)23-s + (0.826 + 0.300i)25-s + (−0.0962 + 0.166i)27-s + (0.346 + 0.412i)29-s + (0.454 + 0.786i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.998350 + 0.681590i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.998350 + 0.681590i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (0.105 - 4.35i)T \) |
good | 5 | \( 1 + (0.134 - 0.764i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.911 - 0.526i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.03 - 1.75i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.81 + 4.99i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.231 + 0.194i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (5.32 - 0.938i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.86 - 2.22i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.52 - 4.38i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.93iT - 37T^{2} \) |
| 41 | \( 1 + (-3.67 - 10.0i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (9.71 + 1.71i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.88 - 8.21i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-9.44 + 1.66i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-4.87 - 4.08i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.68 + 9.55i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.02 + 1.69i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0431 + 0.244i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (4.90 - 1.78i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.40 + 0.511i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.61 + 4.39i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.89 - 16.1i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.49 + 10.1i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−10.25258008378545473248473683997, −9.645771123248366494931548848704, −8.566066577118968002805386647322, −7.71600430431033056549191146966, −6.62255703130850668703748386692, −6.04257643536292749230388658394, −5.04628983419427885429821222625, −3.91050612794630492446575279788, −2.98869400889379451738543961829, −1.28981989245674677075419389321,
0.70876462692010249406105965857, 2.14810611250677050835182856657, 3.80058987173319821705853640037, 4.47777914355668226041987762714, 5.71636199243275387843883782730, 6.56342903452887191932099235243, 7.08743647694139977644118834692, 8.471090563012596298454585954986, 8.994079096551393982468758515581, 9.950227005412818526170727309764