| L(s) = 1 | + (−2.66 − 1.06i)2-s + (−1.72 − 0.187i)3-s + (3.05 + 2.89i)4-s + (−5.63 + 6.62i)5-s + (4.38 + 2.32i)6-s + (−9.66 − 5.81i)7-s + (−0.248 − 0.537i)8-s + (2.92 + 0.644i)9-s + (22.0 − 11.6i)10-s + (3.71 + 0.201i)11-s + (−4.71 − 5.55i)12-s + (2.13 + 9.71i)13-s + (19.5 + 25.7i)14-s + (10.9 − 10.3i)15-s + (−0.819 − 15.1i)16-s + (−9.52 + 5.73i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 − 0.530i)2-s + (−0.573 − 0.0624i)3-s + (0.763 + 0.723i)4-s + (−1.12 + 1.32i)5-s + (0.730 + 0.387i)6-s + (−1.38 − 0.830i)7-s + (−0.0310 − 0.0671i)8-s + (0.325 + 0.0716i)9-s + (2.20 − 1.16i)10-s + (0.337 + 0.0182i)11-s + (−0.393 − 0.462i)12-s + (0.164 + 0.747i)13-s + (1.39 + 1.83i)14-s + (0.729 − 0.690i)15-s + (−0.0511 − 0.944i)16-s + (−0.560 + 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.241261 - 0.146623i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.241261 - 0.146623i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.187i)T \) |
| 59 | \( 1 + (-45.9 + 36.9i)T \) |
| good | 2 | \( 1 + (2.66 + 1.06i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (5.63 - 6.62i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (9.66 + 5.81i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (-3.71 - 0.201i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (-2.13 - 9.71i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (9.52 - 5.73i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (-5.29 + 19.0i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (-14.2 + 9.64i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (6.46 + 16.2i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (-39.7 + 11.0i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (13.7 - 29.7i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (0.305 - 0.451i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (-32.0 + 1.73i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (65.4 - 55.6i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (-46.2 + 87.1i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (-69.3 - 27.6i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (-8.65 - 18.7i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (-26.8 - 31.5i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (-56.1 - 73.9i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (35.8 - 3.89i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (5.57 + 16.5i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (91.5 - 36.4i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (-108. + 143. i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−11.58533847216823882115162487380, −11.25806694566540697797968555077, −10.29882038434265667643993944549, −9.607895103627281989518103094194, −8.248219783813892272366305918887, −6.88322342139251615553437161289, −6.76428367903235530429842873896, −4.15662779986254195266002491112, −2.84961147942512318589966227617, −0.44570639189148798813172222742,
0.798932028216010646269557410758, 3.73005899731475587122445411787, 5.34699918435630016228709051785, 6.55177627847059621491529939715, 7.69921815723166522465805766773, 8.718040512276075991127793436237, 9.287299995540804971802316707251, 10.32544861005420078481897766732, 11.68729245255422346974722857658, 12.48561977381254972968693103040
