L(s) = 1 | + (0.0741 + 0.154i)2-s + (−0.879 + 0.701i)3-s + (1.22 − 1.54i)4-s + (−2.54 + 1.22i)5-s + (−0.173 − 0.0834i)6-s + (−1.82 − 2.28i)7-s + (0.662 + 0.151i)8-s + (−0.386 + 1.69i)9-s + (−0.378 − 0.301i)10-s + (3.89 − 0.888i)11-s + 2.21i·12-s + (0.625 + 2.74i)13-s + (0.217 − 0.450i)14-s + (1.37 − 2.86i)15-s + (−0.851 − 3.72i)16-s − 0.482i·17-s + ⋯ |
L(s) = 1 | + (0.0524 + 0.108i)2-s + (−0.507 + 0.404i)3-s + (0.614 − 0.770i)4-s + (−1.13 + 0.548i)5-s + (−0.0707 − 0.0340i)6-s + (−0.689 − 0.864i)7-s + (0.234 + 0.0534i)8-s + (−0.128 + 0.564i)9-s + (−0.119 − 0.0953i)10-s + (1.17 − 0.267i)11-s + 0.639i·12-s + (0.173 + 0.760i)13-s + (0.0580 − 0.120i)14-s + (0.356 − 0.739i)15-s + (−0.212 − 0.932i)16-s − 0.117i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.612546 + 0.0508684i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.612546 + 0.0508684i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (4.99 - 2.00i)T \) |
good | 2 | \( 1 + (-0.0741 - 0.154i)T + (-1.24 + 1.56i)T^{2} \) |
| 3 | \( 1 + (0.879 - 0.701i)T + (0.667 - 2.92i)T^{2} \) |
| 5 | \( 1 + (2.54 - 1.22i)T + (3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (1.82 + 2.28i)T + (-1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (-3.89 + 0.888i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.625 - 2.74i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + 0.482iT - 17T^{2} \) |
| 19 | \( 1 + (-2.38 - 1.90i)T + (4.22 + 18.5i)T^{2} \) |
| 23 | \( 1 + (4.96 + 2.39i)T + (14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (1.67 + 3.47i)T + (-19.3 + 24.2i)T^{2} \) |
| 37 | \( 1 + (-11.2 - 2.56i)T + (33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + 5.10iT - 41T^{2} \) |
| 43 | \( 1 + (3.56 - 7.40i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-2.32 + 0.531i)T + (42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.401 + 0.193i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 - 1.24T + 59T^{2} \) |
| 61 | \( 1 + (6.71 - 5.35i)T + (13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 + (0.210 - 0.921i)T + (-60.3 - 29.0i)T^{2} \) |
| 71 | \( 1 + (-1.33 - 5.82i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.209 + 0.435i)T + (-45.5 - 57.0i)T^{2} \) |
| 79 | \( 1 + (-1.80 - 0.411i)T + (71.1 + 34.2i)T^{2} \) |
| 83 | \( 1 + (-2.71 + 3.40i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-6.75 - 14.0i)T + (-55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-1.88 - 1.50i)T + (21.5 + 94.5i)T^{2} \) |
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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
−16.52644638739892853264572663897, −16.34615609945955921773230442333, −14.92651124464482794326059335616, −13.87369722205306713180644618555, −11.70359812828073884696619245259, −11.04767643623673313591553656124, −9.815207748564713566227809405726, −7.47509254722675608135405081561, −6.23380234083278213348474027600, −4.04433579375012822567938171473,
3.64588921002318162183508342745, 6.20632046541944219261412287523, 7.66701423508775556807818324872, 9.135913226285766428805878181276, 11.50840114469240598698334959006, 12.07050477197027516797568639871, 12.86014555705453624588421110649, 15.15341468130861380121988456841, 16.00260433388856971168193768160, 17.05033589777420874404518799515