| L(s) = 1 | + (2.38 + 0.639i)2-s + (4.25 + 7.37i)3-s + (−8.57 − 4.94i)4-s + (16.9 − 16.9i)5-s + (5.44 + 20.3i)6-s + (−50.2 + 13.4i)7-s + (−45.2 − 45.2i)8-s + (4.25 − 7.36i)9-s + (51.3 − 29.6i)10-s + (−23.3 + 87.2i)11-s − 84.2i·12-s + (136. + 99.8i)13-s − 128.·14-s + (197. + 52.9i)15-s + (0.133 + 0.231i)16-s + (218. + 126. i)17-s + ⋯ |
| L(s) = 1 | + (0.596 + 0.159i)2-s + (0.473 + 0.819i)3-s + (−0.535 − 0.309i)4-s + (0.678 − 0.678i)5-s + (0.151 + 0.564i)6-s + (−1.02 + 0.274i)7-s + (−0.706 − 0.706i)8-s + (0.0525 − 0.0909i)9-s + (0.513 − 0.296i)10-s + (−0.193 + 0.721i)11-s − 0.585i·12-s + (0.806 + 0.591i)13-s − 0.655·14-s + (0.877 + 0.235i)15-s + (0.000523 + 0.000906i)16-s + (0.755 + 0.436i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.43228 + 0.282135i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43228 + 0.282135i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-136. - 99.8i)T \) |
| good | 2 | \( 1 + (-2.38 - 0.639i)T + (13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (-4.25 - 7.37i)T + (-40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-16.9 + 16.9i)T - 625iT^{2} \) |
| 7 | \( 1 + (50.2 - 13.4i)T + (2.07e3 - 1.20e3i)T^{2} \) |
| 11 | \( 1 + (23.3 - 87.2i)T + (-1.26e4 - 7.32e3i)T^{2} \) |
| 17 | \( 1 + (-218. - 126. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (91.9 + 343. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (735. - 424. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (443. + 768. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-407. + 407. i)T - 9.23e5iT^{2} \) |
| 37 | \( 1 + (-687. + 2.56e3i)T + (-1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + (750. + 201. i)T + (2.44e6 + 1.41e6i)T^{2} \) |
| 43 | \( 1 + (-2.10e3 - 1.21e3i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (858. + 858. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 - 39.2T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-1.14e3 + 306. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (2.41e3 - 4.17e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.51e3 + 943. i)T + (1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 + (165. + 616. i)T + (-2.20e7 + 1.27e7i)T^{2} \) |
| 73 | \( 1 + (-1.00e3 - 1.00e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 3.69e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (3.62e3 - 3.62e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (770. - 2.87e3i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-28.3 - 105. i)T + (-7.66e7 + 4.42e7i)T^{2} \) |
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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
−19.41263946105138196910142955250, −17.87958924268940985090939379778, −16.13380729086723131707023051647, −15.11503513810224694817742772271, −13.67289451048301845053457948527, −12.62508852800930288508717967732, −9.807821865825716835253243506171, −9.193004068888824594543117900167, −5.91852614200448930082295604907, −4.05125981497417476349102787578,
3.13524797595248404695666079104, 6.16269949777354512179855304742, 8.197661304356482273539222299240, 10.20989237222476957460907444540, 12.49817903215799582521311872492, 13.54166518722480980676938714998, 14.21952740129029701607126246856, 16.39636441627047616071940318699, 18.19688169879828461564933799097, 18.81947914922791570393112100725
