| L(s) = 1 | + (0.488 − 0.776i)2-s + (−1.66 − 4.74i)3-s + (1.37 + 2.84i)4-s + (−1.98 + 0.453i)5-s + (−4.50 − 1.02i)6-s + (9.56 + 4.60i)7-s + (6.52 + 0.735i)8-s + (−12.7 + 10.1i)9-s + (−0.616 + 1.76i)10-s + (−11.6 + 1.31i)11-s + (11.2 − 11.2i)12-s + (−11.0 − 8.79i)13-s + (8.24 − 5.18i)14-s + (5.45 + 8.67i)15-s + (−4.12 + 5.16i)16-s + (0.154 + 0.154i)17-s + ⋯ |
| L(s) = 1 | + (0.244 − 0.388i)2-s + (−0.553 − 1.58i)3-s + (0.342 + 0.711i)4-s + (−0.396 + 0.0906i)5-s + (−0.750 − 0.171i)6-s + (1.36 + 0.657i)7-s + (0.815 + 0.0919i)8-s + (−1.41 + 1.13i)9-s + (−0.0616 + 0.176i)10-s + (−1.05 + 0.119i)11-s + (0.936 − 0.936i)12-s + (−0.848 − 0.676i)13-s + (0.588 − 0.370i)14-s + (0.363 + 0.578i)15-s + (−0.257 + 0.323i)16-s + (0.00910 + 0.00910i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.859544 - 0.519135i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.859544 - 0.519135i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (15.7 + 24.3i)T \) |
| good | 2 | \( 1 + (-0.488 + 0.776i)T + (-1.73 - 3.60i)T^{2} \) |
| 3 | \( 1 + (1.66 + 4.74i)T + (-7.03 + 5.61i)T^{2} \) |
| 5 | \( 1 + (1.98 - 0.453i)T + (22.5 - 10.8i)T^{2} \) |
| 7 | \( 1 + (-9.56 - 4.60i)T + (30.5 + 38.3i)T^{2} \) |
| 11 | \( 1 + (11.6 - 1.31i)T + (117. - 26.9i)T^{2} \) |
| 13 | \( 1 + (11.0 + 8.79i)T + (37.6 + 164. i)T^{2} \) |
| 17 | \( 1 + (-0.154 - 0.154i)T + 289iT^{2} \) |
| 19 | \( 1 + (-20.3 - 7.12i)T + (282. + 225. i)T^{2} \) |
| 23 | \( 1 + (-1.51 + 6.62i)T + (-476. - 229. i)T^{2} \) |
| 31 | \( 1 + (0.406 - 0.646i)T + (-416. - 865. i)T^{2} \) |
| 37 | \( 1 + (6.96 + 0.784i)T + (1.33e3 + 304. i)T^{2} \) |
| 41 | \( 1 + (-35.8 + 35.8i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-18.1 + 11.4i)T + (802. - 1.66e3i)T^{2} \) |
| 47 | \( 1 + (2.57 + 22.8i)T + (-2.15e3 + 491. i)T^{2} \) |
| 53 | \( 1 + (-12.7 - 55.8i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 + 48.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + (24.4 + 69.8i)T + (-2.90e3 + 2.32e3i)T^{2} \) |
| 67 | \( 1 + (-33.7 + 26.9i)T + (998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (-78.3 - 62.5i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (-5.59 - 8.90i)T + (-2.31e3 + 4.80e3i)T^{2} \) |
| 79 | \( 1 + (-4.65 + 41.2i)T + (-6.08e3 - 1.38e3i)T^{2} \) |
| 83 | \( 1 + (-85.5 + 41.1i)T + (4.29e3 - 5.38e3i)T^{2} \) |
| 89 | \( 1 + (89.2 - 142. i)T + (-3.43e3 - 7.13e3i)T^{2} \) |
| 97 | \( 1 + (-20.2 + 57.9i)T + (-7.35e3 - 5.86e3i)T^{2} \) |
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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
−17.16850074450450267801836593695, −15.47498200219575080012435292779, −13.83770272992953111112053281161, −12.59170739203885546360805801045, −11.93698043641229834294827352861, −11.01946568590702735627916149109, −7.914172458501602068251131731395, −7.56610913536133841216382978870, −5.37205647845159185847971192468, −2.26853040547057054155404056923,
4.52504164191194408946065898757, 5.33475322398810176268929386281, 7.58632643514464683914865254181, 9.686240590628959892687459855479, 10.81102061500491318263156519581, 11.50581895660908274251584403118, 14.03669515958099033294414169551, 14.93922157394777123780629646483, 15.86031556613409573045068230384, 16.67361498564826132023047509789
