L(s) = 1 | + (−41.7 + 17.5i)2-s + (364. + 210. i)3-s + (1.43e3 − 1.46e3i)4-s + 505. i·5-s + (−1.88e4 − 2.38e3i)6-s − 7.75e4i·7-s + (−3.38e4 + 8.62e4i)8-s + (8.82e4 + 1.53e5i)9-s + (−8.88e3 − 2.10e4i)10-s + 7.31e5·11-s + (8.30e5 − 2.32e5i)12-s + 1.14e6·13-s + (1.36e6 + 3.23e6i)14-s + (−1.06e5 + 1.84e5i)15-s + (−1.03e5 − 4.19e6i)16-s + 3.04e6i·17-s + ⋯ |
L(s) = 1 | + (−0.921 + 0.388i)2-s + (0.865 + 0.500i)3-s + (0.698 − 0.715i)4-s + 0.0723i·5-s + (−0.992 − 0.125i)6-s − 1.74i·7-s + (−0.365 + 0.930i)8-s + (0.498 + 0.866i)9-s + (−0.0280 − 0.0666i)10-s + 1.36·11-s + (0.962 − 0.269i)12-s + 0.853·13-s + (0.677 + 1.60i)14-s + (−0.0362 + 0.0625i)15-s + (−0.0247 − 0.999i)16-s + 0.519i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(1.62462 + 0.223339i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62462 + 0.223339i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (41.7 - 17.5i)T \) |
| 3 | \( 1 + (-364. - 210. i)T \) |
good | 5 | \( 1 - 505. iT - 4.88e7T^{2} \) |
| 7 | \( 1 + 7.75e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 - 7.31e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.14e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.04e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 4.99e6iT - 1.16e14T^{2} \) |
| 23 | \( 1 - 5.14e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.44e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 + 2.51e7iT - 2.54e16T^{2} \) |
| 37 | \( 1 + 1.69e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 3.25e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + 1.17e9iT - 9.29e17T^{2} \) |
| 47 | \( 1 + 1.82e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.85e7iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 5.59e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 8.08e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.71e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 1.89e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.88e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.26e10iT - 7.47e20T^{2} \) |
| 83 | \( 1 + 3.76e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 5.79e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 9.56e10T + 7.15e21T^{2} \) |
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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
−17.22835133600602362966098933034, −16.35346870566908142315926056603, −14.78117250553054288150859195790, −13.74616179247402510354779224119, −10.95771354007402067094799586534, −9.812005193857193280010660107732, −8.319278371909944962969518598584, −6.83967918590782526722180832005, −3.89660881132630157398980338784, −1.25737558067352654367529404567,
1.47951982423957278193418649219, 3.04922691432557963364513434730, 6.63263909970939395996462747342, 8.618630848225472385813412251780, 9.222117319687313251871707789049, 11.60167956480674652430819381769, 12.71436040144732391250115822201, 14.71712542863246213359675681845, 16.03363762822770361016271814712, 17.92065539434707032255699574229