| L(s) = 1 | + (−1.31 − 3.34i)2-s + (0.344 + 4.59i)3-s + (−3.62 + 3.36i)4-s + (−16.7 + 11.4i)5-s + (14.9 − 7.19i)6-s + (−11.8 + 14.2i)7-s + (−9.90 − 4.77i)8-s + (5.71 − 0.861i)9-s + (60.3 + 41.1i)10-s + (8.20 + 1.23i)11-s + (−16.6 − 15.4i)12-s + (−1.90 + 2.38i)13-s + (63.1 + 21.1i)14-s + (−58.3 − 73.1i)15-s + (−5.91 + 78.8i)16-s + (−6.40 + 1.97i)17-s + ⋯ |
| L(s) = 1 | + (−0.464 − 1.18i)2-s + (0.0662 + 0.884i)3-s + (−0.453 + 0.420i)4-s + (−1.50 + 1.02i)5-s + (1.01 − 0.489i)6-s + (−0.641 + 0.767i)7-s + (−0.437 − 0.210i)8-s + (0.211 − 0.0319i)9-s + (1.90 + 1.30i)10-s + (0.224 + 0.0339i)11-s + (−0.401 − 0.372i)12-s + (−0.0405 + 0.0508i)13-s + (1.20 + 0.403i)14-s + (−1.00 − 1.25i)15-s + (−0.0923 + 1.23i)16-s + (−0.0914 + 0.0282i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0492 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0492 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.333147 + 0.349993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.333147 + 0.349993i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (11.8 - 14.2i)T \) |
| good | 2 | \( 1 + (1.31 + 3.34i)T + (-5.86 + 5.44i)T^{2} \) |
| 3 | \( 1 + (-0.344 - 4.59i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (16.7 - 11.4i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-8.20 - 1.23i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (1.90 - 2.38i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (6.40 - 1.97i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-4.06 + 7.04i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (191. + 59.1i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (19.9 - 87.5i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-159. - 275. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-175. - 163. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (32.0 + 15.4i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-116. + 56.3i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (68.8 + 175. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (191. - 177. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-364. - 248. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (566. + 525. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-267. - 462. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-192. - 843. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (331. - 845. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-395. + 684. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (35.5 + 44.5i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-416. + 62.7i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + 1.61e3T + 9.12e5T^{2} \) |
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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
−15.58019856147339674453028304953, −14.59798464124615298315843502371, −12.43226704597419562969216578006, −11.72103382844965128773342154560, −10.63387737767543274665759759832, −9.844336303027356164937141144212, −8.491397783358043402194208898330, −6.66199113454117800878298497803, −4.03453418988464557686850741180, −2.95943760696418905589226822476,
0.42004397336755907633541027702, 4.15595189048705550931489336660, 6.29946723188781913901592602085, 7.67875163846285106290014673325, 7.921617548972518061325553652623, 9.505038023090519139894085689806, 11.66437817572562750624265477732, 12.54756960592721513671393178382, 13.69217150529919800078325069869, 15.29273622606716926924797471145
