| L(s) = 1 | + (0.460 + 0.531i)2-s + (0.841 + 0.540i)3-s + (0.214 − 1.48i)4-s + (−0.700 + 1.53i)5-s + (0.100 + 0.696i)6-s + (−3.11 − 0.915i)7-s + (2.07 − 1.33i)8-s + (0.415 + 0.909i)9-s + (−1.13 + 0.334i)10-s + (−1.33 + 1.54i)11-s + (0.985 − 1.13i)12-s + (−0.227 + 0.0666i)13-s + (−0.949 − 2.07i)14-s + (−1.41 + 0.911i)15-s + (−1.22 − 0.358i)16-s + (−0.427 − 2.97i)17-s + ⋯ |
| L(s) = 1 | + (0.325 + 0.376i)2-s + (0.485 + 0.312i)3-s + (0.107 − 0.744i)4-s + (−0.313 + 0.685i)5-s + (0.0408 + 0.284i)6-s + (−1.17 − 0.345i)7-s + (0.733 − 0.471i)8-s + (0.138 + 0.303i)9-s + (−0.360 + 0.105i)10-s + (−0.403 + 0.465i)11-s + (0.284 − 0.328i)12-s + (−0.0629 + 0.0184i)13-s + (−0.253 − 0.555i)14-s + (−0.366 + 0.235i)15-s + (−0.305 − 0.0897i)16-s + (−0.103 − 0.721i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07366 + 0.228706i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07366 + 0.228706i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (0.779 - 4.73i)T \) |
| good | 2 | \( 1 + (-0.460 - 0.531i)T + (-0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (0.700 - 1.53i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (3.11 + 0.915i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (1.33 - 1.54i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.227 - 0.0666i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.427 + 2.97i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.682 + 4.74i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.01 - 7.03i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.64 + 3.62i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-4.77 - 10.4i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.40 + 7.46i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.42 - 1.55i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + (0.510 + 0.149i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-3.30 + 0.971i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.08 + 2.62i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (9.77 + 11.2i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (8.80 + 10.1i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.632 - 4.39i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (2.00 - 0.589i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.67 - 5.86i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.11 - 1.35i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.753 + 1.65i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−14.99939083328353249593347106194, −13.83719242240788324342092449307, −13.08726598291729458493724204189, −11.29960214123430266438089453971, −10.16690323290249516944898186178, −9.404567210645883410706753140300, −7.40369191038476995262195890778, −6.54996171851442874928320178134, −4.83935539443506707363986222571, −3.09069568170806516868877563379,
2.78709468377455288976040547982, 4.16850659623982714120888986580, 6.23083740558236525576155175982, 7.88746319439776089319809899125, 8.692815298706124807089927143493, 10.16855272946492715256362648671, 11.78262596271872954598083991525, 12.71692149508787005864476144397, 13.12100652589629072363530532276, 14.47201583529838882406086811944
