| L(s) = 1 | + (0.283 + 0.111i)2-s + (−4.60 − 2.41i)3-s + (−5.79 − 5.37i)4-s + (14.9 + 10.2i)5-s + (−1.03 − 1.19i)6-s + (−13.0 + 13.1i)7-s + (−2.09 − 4.35i)8-s + (15.3 + 22.2i)9-s + (3.10 + 4.55i)10-s + (−4.06 − 26.9i)11-s + (13.6 + 38.7i)12-s + (40.2 − 32.1i)13-s + (−5.14 + 2.28i)14-s + (−44.2 − 83.1i)15-s + (4.61 + 61.6i)16-s + (81.2 + 25.0i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.0392i)2-s + (−0.885 − 0.465i)3-s + (−0.724 − 0.672i)4-s + (1.33 + 0.913i)5-s + (−0.0703 − 0.0813i)6-s + (−0.702 + 0.711i)7-s + (−0.0927 − 0.192i)8-s + (0.567 + 0.823i)9-s + (0.0982 + 0.144i)10-s + (−0.111 − 0.738i)11-s + (0.328 + 0.932i)12-s + (0.859 − 0.685i)13-s + (−0.0982 + 0.0436i)14-s + (−0.761 − 1.43i)15-s + (0.0721 + 0.962i)16-s + (1.15 + 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.35635 - 0.131161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35635 - 0.131161i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.60 + 2.41i)T \) |
| 7 | \( 1 + (13.0 - 13.1i)T \) |
| good | 2 | \( 1 + (-0.283 - 0.111i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-14.9 - 10.2i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (4.06 + 26.9i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-40.2 + 32.1i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-81.2 - 25.0i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-94.1 + 54.3i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (8.40 + 27.2i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-251. + 57.3i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-33.3 - 19.2i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (84.6 - 78.5i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (416. - 200. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-296. - 143. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (67.7 - 172. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-200. + 215. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-433. + 295. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-467. - 503. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (123. - 213. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (306. + 70.0i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (263. - 103. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-52.8 - 91.5i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (280. - 352. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (1.02e3 + 155. i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 + 214. iT - 9.12e5T^{2} \) |
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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
−12.81565668223319577874280757219, −11.47271973289269790104920696822, −10.24249993730846232531067753752, −9.944308114298720556310950749406, −8.483684862221818643148297096165, −6.63686926137944411756224117217, −5.90342800331336392743113761097, −5.34846320673357789190502531155, −2.98995828000366056161740662044, −1.07790496497885169981107689790,
1.05143258939320492847489903986, 3.64714904169726314073492113130, 4.86533675701954660410616189918, 5.76351271548647262482256291837, 7.11207372930922577796800342723, 8.786750085769800606851774603463, 9.763947636530014863865719466286, 10.16387751887592738610436870178, 11.96577271586447671749506398616, 12.54710838911248945572684319296
