| L(s) = 1 | + 5.95i·2-s + (−43.1 + 18.1i)3-s + 92.4·4-s + (−108. − 256. i)6-s + 1.31e3i·8-s + (1.52e3 − 1.56e3i)9-s + (−3.98e3 + 1.67e3i)12-s + 1.34e4·13-s + 4.01e3·16-s + (9.31e3 + 9.11e3i)18-s + 5.83e4i·23-s + (−2.38e4 − 5.66e4i)24-s − 7.81e4·25-s + 8.03e4i·26-s + (−3.75e4 + 9.51e4i)27-s + ⋯ |
| L(s) = 1 | + 0.526i·2-s + (−0.921 + 0.387i)3-s + 0.722·4-s + (−0.204 − 0.485i)6-s + 0.907i·8-s + (0.699 − 0.714i)9-s + (−0.666 + 0.280i)12-s + 1.70·13-s + 0.244·16-s + (0.376 + 0.368i)18-s + 0.999i·23-s + (−0.351 − 0.836i)24-s − 25-s + 0.896i·26-s + (−0.367 + 0.930i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.933896 + 1.40618i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.933896 + 1.40618i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (43.1 - 18.1i)T \) |
| 23 | \( 1 - 5.83e4iT \) |
| good | 2 | \( 1 - 5.95iT - 128T^{2} \) |
| 5 | \( 1 + 7.81e4T^{2} \) |
| 7 | \( 1 - 8.23e5T^{2} \) |
| 11 | \( 1 + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.34e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.10e8T^{2} \) |
| 19 | \( 1 - 8.93e8T^{2} \) |
| 29 | \( 1 - 2.26e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 9.49e10T^{2} \) |
| 41 | \( 1 + 3.51e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 2.71e11T^{2} \) |
| 47 | \( 1 - 9.14e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.15e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 3.14e12T^{2} \) |
| 67 | \( 1 - 6.06e12T^{2} \) |
| 71 | \( 1 + 6.00e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 6.03e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.92e13T^{2} \) |
| 83 | \( 1 + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.07e13T^{2} \) |
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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
−13.70783197353329276687957785585, −12.30314110282110740257964564065, −11.24222569904597914683191030205, −10.60393777825453817184769452681, −9.004779015569574134611819677017, −7.47063120436727318376556878987, −6.27981283412094986240625700330, −5.43933505534388875115507208415, −3.64464443365419623857976543273, −1.40456869941852292731775402410,
0.71647984118915061397246067625, 2.03921351651026882983297755315, 3.91688134892640730184524312859, 5.83654372548428280148044240760, 6.69588572555970461069963158561, 8.097050632847916522626087719113, 9.956918890340273170978113143292, 11.00535005195954951279999951998, 11.62191244607868156024553943103, 12.71463757042706322173028345749
