| L(s) = 1 | + (−0.240 − 1.05i)2-s + (53.2 + 25.6i)3-s + (114. − 55.0i)4-s + (117. + 516. i)5-s + (14.1 − 62.1i)6-s + (−631. − 304. i)7-s + (−171. − 215. i)8-s + (814. + 1.02e3i)9-s + (514. − 247. i)10-s + (356. − 446. i)11-s + 7.49e3·12-s + (−5.19e3 + 6.50e3i)13-s + (−168. + 737. i)14-s + (−6.96e3 + 3.05e4i)15-s + (9.93e3 − 1.24e4i)16-s + 1.94e4·17-s + ⋯ |
| L(s) = 1 | + (−0.0212 − 0.0930i)2-s + (1.13 + 0.548i)3-s + (0.892 − 0.429i)4-s + (0.421 + 1.84i)5-s + (0.0268 − 0.117i)6-s + (−0.696 − 0.335i)7-s + (−0.118 − 0.148i)8-s + (0.372 + 0.467i)9-s + (0.162 − 0.0784i)10-s + (0.0807 − 0.101i)11-s + 1.25·12-s + (−0.655 + 0.821i)13-s + (−0.0164 + 0.0718i)14-s + (−0.532 + 2.33i)15-s + (0.606 − 0.760i)16-s + 0.959·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.783i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.620 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.52859 + 1.22297i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52859 + 1.22297i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-1.18e5 - 5.61e4i)T \) |
| good | 2 | \( 1 + (0.240 + 1.05i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (-53.2 - 25.6i)T + (1.36e3 + 1.70e3i)T^{2} \) |
| 5 | \( 1 + (-117. - 516. i)T + (-7.03e4 + 3.38e4i)T^{2} \) |
| 7 | \( 1 + (631. + 304. i)T + (5.13e5 + 6.43e5i)T^{2} \) |
| 11 | \( 1 + (-356. + 446. i)T + (-4.33e6 - 1.89e7i)T^{2} \) |
| 13 | \( 1 + (5.19e3 - 6.50e3i)T + (-1.39e7 - 6.11e7i)T^{2} \) |
| 17 | \( 1 - 1.94e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + (-2.18e4 + 1.05e4i)T + (5.57e8 - 6.98e8i)T^{2} \) |
| 23 | \( 1 + (-1.09e4 + 4.78e4i)T + (-3.06e9 - 1.47e9i)T^{2} \) |
| 31 | \( 1 + (6.03e4 + 2.64e5i)T + (-2.47e10 + 1.19e10i)T^{2} \) |
| 37 | \( 1 + (-1.33e5 - 1.66e5i)T + (-2.11e10 + 9.25e10i)T^{2} \) |
| 41 | \( 1 + 3.67e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + (8.32e4 - 3.64e5i)T + (-2.44e11 - 1.17e11i)T^{2} \) |
| 47 | \( 1 + (-5.76e5 + 7.22e5i)T + (-1.12e11 - 4.93e11i)T^{2} \) |
| 53 | \( 1 + (7.62e3 + 3.34e4i)T + (-1.05e12 + 5.09e11i)T^{2} \) |
| 59 | \( 1 + 2.88e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-2.22e6 - 1.07e6i)T + (1.95e12 + 2.45e12i)T^{2} \) |
| 67 | \( 1 + (1.15e6 + 1.44e6i)T + (-1.34e12 + 5.90e12i)T^{2} \) |
| 71 | \( 1 + (-1.24e5 + 1.56e5i)T + (-2.02e12 - 8.86e12i)T^{2} \) |
| 73 | \( 1 + (2.49e5 - 1.09e6i)T + (-9.95e12 - 4.79e12i)T^{2} \) |
| 79 | \( 1 + (1.82e6 + 2.29e6i)T + (-4.27e12 + 1.87e13i)T^{2} \) |
| 83 | \( 1 + (2.44e6 - 1.17e6i)T + (1.69e13 - 2.12e13i)T^{2} \) |
| 89 | \( 1 + (1.87e5 + 8.19e5i)T + (-3.98e13 + 1.91e13i)T^{2} \) |
| 97 | \( 1 + (1.24e7 - 5.97e6i)T + (5.03e13 - 6.31e13i)T^{2} \) |
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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
−15.29326278932341070231366824877, −14.59650496081093755314273247424, −13.85505262889827113778067578228, −11.61030518231306808867030199682, −10.25421718105623916651265080241, −9.668852997945725969547314498636, −7.38051224145245865409957624962, −6.35340350251480609953212475513, −3.33192773280331485321948031265, −2.46501834623076685213722929416,
1.45297951208783718196517966415, 3.05447588022449639682207387486, 5.54277337856601492844088165412, 7.55738409490402076826144521097, 8.545815982029818112971200147345, 9.731156135123607876193214157906, 12.26000718293917273372257349646, 12.69674189259784102963307885463, 13.94810639974255436684726025036, 15.56039158826337321320494642475
