| L(s) = 1 | − 45.2i·2-s + (385. + 618. i)3-s − 2.04e3·4-s + 1.63e4i·5-s + (2.79e4 − 1.74e4i)6-s + 2.13e5·7-s + 9.26e4i·8-s + (−2.34e5 + 4.77e5i)9-s + 7.39e5·10-s + 852. i·11-s + (−7.89e5 − 1.26e6i)12-s − 3.33e6·13-s − 9.64e6i·14-s + (−1.01e7 + 6.30e6i)15-s + 4.19e6·16-s + 4.86e6i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.528 + 0.848i)3-s − 0.500·4-s + 1.04i·5-s + (0.600 − 0.374i)6-s + 1.81·7-s + 0.353i·8-s + (−0.440 + 0.897i)9-s + 0.739·10-s + 0.000481i·11-s + (−0.264 − 0.424i)12-s − 0.690·13-s − 1.28i·14-s + (−0.887 + 0.553i)15-s + 0.250·16-s + 0.201i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.82152 + 0.521162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82152 + 0.521162i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 45.2iT \) |
| 3 | \( 1 + (-385. - 618. i)T \) |
| good | 5 | \( 1 - 1.63e4iT - 2.44e8T^{2} \) |
| 7 | \( 1 - 2.13e5T + 1.38e10T^{2} \) |
| 11 | \( 1 - 852. iT - 3.13e12T^{2} \) |
| 13 | \( 1 + 3.33e6T + 2.32e13T^{2} \) |
| 17 | \( 1 - 4.86e6iT - 5.82e14T^{2} \) |
| 19 | \( 1 - 1.39e5T + 2.21e15T^{2} \) |
| 23 | \( 1 + 2.14e8iT - 2.19e16T^{2} \) |
| 29 | \( 1 + 8.70e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 - 1.12e9T + 7.87e17T^{2} \) |
| 37 | \( 1 - 1.03e8T + 6.58e18T^{2} \) |
| 41 | \( 1 + 6.26e8iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 5.27e9T + 3.99e19T^{2} \) |
| 47 | \( 1 - 1.16e10iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 2.31e10iT - 4.91e20T^{2} \) |
| 59 | \( 1 + 3.43e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 + 1.42e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + 4.78e10T + 8.18e21T^{2} \) |
| 71 | \( 1 - 2.46e11iT - 1.64e22T^{2} \) |
| 73 | \( 1 - 5.45e9T + 2.29e22T^{2} \) |
| 79 | \( 1 + 3.27e9T + 5.90e22T^{2} \) |
| 83 | \( 1 + 1.55e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 - 2.37e10iT - 2.46e23T^{2} \) |
| 97 | \( 1 - 7.71e11T + 6.93e23T^{2} \) |
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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
−20.52007013971099503107923404373, −18.90188626600168903086708429505, −17.41273698192477745823493911977, −14.95709231531537142669963541456, −14.17193156421105519318947966466, −11.42638366898050964808187764095, −10.27260477188170514515708973493, −8.179749050940782613802923553780, −4.57867095536885444855804477088, −2.43520548979333277248717840343,
1.33983742148211854761494666971, 5.02441812133786742099649360934, 7.61431397023134275482323105549, 8.790394794587510301183306555561, 12.01373552596451827381966293520, 13.69761656897018926542940204989, 14.97607147414274186435781254509, 17.09008960730660097679570296307, 18.11526957541138320467464301455, 19.96614783775376513956433929054
