| L(s) = 1 | + (−1.73 − 7.58i)2-s + (−20.8 − 10.0i)3-s + (−25.6 + 12.3i)4-s + (−4.14 − 18.1i)5-s + (−40.0 + 175. i)6-s + (89.9 + 43.3i)7-s + (−16.9 − 21.2i)8-s + (182. + 228. i)9-s + (−130. + 62.8i)10-s + (−167. + 209. i)11-s + 659.·12-s + (122. − 153. i)13-s + (172. − 756. i)14-s + (−95.8 + 420. i)15-s + (−700. + 878. i)16-s − 2.30e3·17-s + ⋯ |
| L(s) = 1 | + (−0.305 − 1.34i)2-s + (−1.33 − 0.644i)3-s + (−0.802 + 0.386i)4-s + (−0.0740 − 0.324i)5-s + (−0.454 + 1.99i)6-s + (0.693 + 0.334i)7-s + (−0.0934 − 0.117i)8-s + (0.751 + 0.941i)9-s + (−0.412 + 0.198i)10-s + (−0.416 + 0.522i)11-s + 1.32·12-s + (0.200 − 0.251i)13-s + (0.235 − 1.03i)14-s + (−0.110 + 0.481i)15-s + (−0.684 + 0.857i)16-s − 1.93·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0639 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0639 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.212243 + 0.226290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.212243 + 0.226290i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (4.33e3 - 1.30e3i)T \) |
| good | 2 | \( 1 + (1.73 + 7.58i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (20.8 + 10.0i)T + (151. + 189. i)T^{2} \) |
| 5 | \( 1 + (4.14 + 18.1i)T + (-2.81e3 + 1.35e3i)T^{2} \) |
| 7 | \( 1 + (-89.9 - 43.3i)T + (1.04e4 + 1.31e4i)T^{2} \) |
| 11 | \( 1 + (167. - 209. i)T + (-3.58e4 - 1.57e5i)T^{2} \) |
| 13 | \( 1 + (-122. + 153. i)T + (-8.26e4 - 3.61e5i)T^{2} \) |
| 17 | \( 1 + 2.30e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (313. - 151. i)T + (1.54e6 - 1.93e6i)T^{2} \) |
| 23 | \( 1 + (-680. + 2.98e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 31 | \( 1 + (-51.7 - 226. i)T + (-2.57e7 + 1.24e7i)T^{2} \) |
| 37 | \( 1 + (2.97e3 + 3.73e3i)T + (-1.54e7 + 6.76e7i)T^{2} \) |
| 41 | \( 1 + 1.20e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (3.25e3 - 1.42e4i)T + (-1.32e8 - 6.37e7i)T^{2} \) |
| 47 | \( 1 + (-1.85e4 + 2.32e4i)T + (-5.10e7 - 2.23e8i)T^{2} \) |
| 53 | \( 1 + (3.11e3 + 1.36e4i)T + (-3.76e8 + 1.81e8i)T^{2} \) |
| 59 | \( 1 - 7.83e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + (4.89e4 + 2.35e4i)T + (5.26e8 + 6.60e8i)T^{2} \) |
| 67 | \( 1 + (1.71e4 + 2.14e4i)T + (-3.00e8 + 1.31e9i)T^{2} \) |
| 71 | \( 1 + (-1.94e4 + 2.43e4i)T + (-4.01e8 - 1.75e9i)T^{2} \) |
| 73 | \( 1 + (-685. + 3.00e3i)T + (-1.86e9 - 8.99e8i)T^{2} \) |
| 79 | \( 1 + (-3.01e4 - 3.78e4i)T + (-6.84e8 + 2.99e9i)T^{2} \) |
| 83 | \( 1 + (7.36e4 - 3.54e4i)T + (2.45e9 - 3.07e9i)T^{2} \) |
| 89 | \( 1 + (-1.41e4 - 6.20e4i)T + (-5.03e9 + 2.42e9i)T^{2} \) |
| 97 | \( 1 + (-1.08e5 + 5.21e4i)T + (5.35e9 - 6.71e9i)T^{2} \) |
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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
−15.31223663553669425063267166845, −13.08454625061918386708911146880, −12.35504487715773877013726095919, −11.29860783831587703460123923668, −10.59527886830858288293092216240, −8.738122874145017823954359018698, −6.64454399282995421170856323319, −4.81649756884279680391197490053, −1.94808318803502884596454907662, −0.24878883674979592251037275891,
4.73258878172335606350106689933, 5.97847360903638319404637096147, 7.23710895827383287715814239022, 8.902446856422777082456022479803, 10.79321414397937127164499234404, 11.48471051520253423727577141616, 13.63698212536567879581142241726, 15.18866978958342960711903255556, 15.83150206182567656367707310934, 17.02437890839863169921310361942
