| L(s) = 1 | + (1.69 − 0.818i)2-s + (5.27 + 6.61i)3-s + (−17.7 + 22.2i)4-s + (−69.8 + 33.6i)5-s + (14.3 + 6.92i)6-s + (10.7 + 13.4i)7-s + (−25.3 + 111. i)8-s + (38.1 − 167. i)9-s + (−91.1 + 114. i)10-s + (117. + 512. i)11-s − 240.·12-s + (61.6 + 270. i)13-s + (29.2 + 14.1i)14-s + (−591. − 284. i)15-s + (−154. − 677. i)16-s + 1.48e3·17-s + ⋯ |
| L(s) = 1 | + (0.300 − 0.144i)2-s + (0.338 + 0.424i)3-s + (−0.554 + 0.694i)4-s + (−1.25 + 0.602i)5-s + (0.163 + 0.0785i)6-s + (0.0829 + 0.103i)7-s + (−0.140 + 0.613i)8-s + (0.156 − 0.687i)9-s + (−0.288 + 0.361i)10-s + (0.291 + 1.27i)11-s − 0.482·12-s + (0.101 + 0.443i)13-s + (0.0399 + 0.0192i)14-s + (−0.678 − 0.326i)15-s + (−0.151 − 0.662i)16-s + 1.24·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.641948 + 1.02797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.641948 + 1.02797i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-3.80e3 - 2.45e3i)T \) |
| good | 2 | \( 1 + (-1.69 + 0.818i)T + (19.9 - 25.0i)T^{2} \) |
| 3 | \( 1 + (-5.27 - 6.61i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (69.8 - 33.6i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + (-10.7 - 13.4i)T + (-3.73e3 + 1.63e4i)T^{2} \) |
| 11 | \( 1 + (-117. - 512. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-61.6 - 270. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 - 1.48e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (231. - 290. i)T + (-5.50e5 - 2.41e6i)T^{2} \) |
| 23 | \( 1 + (1.78e3 + 860. i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 31 | \( 1 + (3.77e3 - 1.81e3i)T + (1.78e7 - 2.23e7i)T^{2} \) |
| 37 | \( 1 + (2.98e3 - 1.30e4i)T + (-6.24e7 - 3.00e7i)T^{2} \) |
| 41 | \( 1 + 2.20e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-3.89e3 - 1.87e3i)T + (9.16e7 + 1.14e8i)T^{2} \) |
| 47 | \( 1 + (-4.69e3 - 2.05e4i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-2.73e4 + 1.31e4i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 - 3.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (2.56e4 + 3.21e4i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (7.81e3 - 3.42e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-8.11e3 - 3.55e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (3.15e4 + 1.51e4i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + (-8.45e3 + 3.70e4i)T + (-2.77e9 - 1.33e9i)T^{2} \) |
| 83 | \( 1 + (-2.37e4 + 2.98e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (4.84e4 - 2.33e4i)T + (3.48e9 - 4.36e9i)T^{2} \) |
| 97 | \( 1 + (-4.05e4 + 5.08e4i)T + (-1.91e9 - 8.37e9i)T^{2} \) |
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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
−16.28166623636805181479938865179, −15.03061313000933902195251350514, −14.31707900570865654524110858420, −12.41417995833873770567903829901, −11.81668146253581996550372722529, −9.942190627580730124880474075230, −8.457789815986758609292286324495, −7.15730199717684045774709654493, −4.40408357640962764778930472631, −3.36923785350875794113669661484,
0.73508821803041543219722648766, 3.90582291410830310852917548448, 5.55632568291552166479168025086, 7.69091513433448056495567376109, 8.743502365996242289149175413330, 10.58502374981045151492790977368, 12.09867722321921102872285423661, 13.38320592109994830315943278833, 14.30320370039904705980227089022, 15.65619068809785770972522396406
