| L(s) = 1 | + (0.819 + 0.573i)2-s + (0.264 + 1.71i)3-s + (0.342 + 0.939i)4-s + (−2.23 − 0.138i)5-s + (−0.765 + 1.55i)6-s + (0.660 + 1.41i)7-s + (−0.258 + 0.965i)8-s + (−2.86 + 0.903i)9-s + (−1.74 − 1.39i)10-s + (2.49 + 2.97i)11-s + (−1.51 + 0.833i)12-s + (−2.53 − 3.62i)13-s + (−0.271 + 1.53i)14-s + (−0.352 − 3.85i)15-s + (−0.766 + 0.642i)16-s + (1.52 + 5.69i)17-s + ⋯ |
| L(s) = 1 | + (0.579 + 0.405i)2-s + (0.152 + 0.988i)3-s + (0.171 + 0.469i)4-s + (−0.998 − 0.0618i)5-s + (−0.312 + 0.634i)6-s + (0.249 + 0.535i)7-s + (−0.0915 + 0.341i)8-s + (−0.953 + 0.301i)9-s + (−0.553 − 0.440i)10-s + (0.753 + 0.897i)11-s + (−0.438 + 0.240i)12-s + (−0.703 − 1.00i)13-s + (−0.0725 + 0.411i)14-s + (−0.0910 − 0.995i)15-s + (−0.191 + 0.160i)16-s + (0.370 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.726040 + 1.33188i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.726040 + 1.33188i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.819 - 0.573i)T \) |
| 3 | \( 1 + (-0.264 - 1.71i)T \) |
| 5 | \( 1 + (2.23 + 0.138i)T \) |
| good | 7 | \( 1 + (-0.660 - 1.41i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-2.49 - 2.97i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.53 + 3.62i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.52 - 5.69i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.43 + 3.13i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.84 + 0.862i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.651 + 3.69i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.48 + 1.63i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-5.27 + 1.41i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.36 - 0.240i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.741 - 8.47i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (2.58 - 1.20i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-4.58 - 4.58i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.286 - 0.240i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.34 - 1.21i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.14 + 6.40i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (13.8 + 7.99i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (15.8 + 4.24i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.515 + 0.0908i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.06 + 8.66i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (2.59 + 4.49i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (11.4 + 1.00i)T + (95.5 + 16.8i)T^{2} \) |
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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
−12.07016996841701875619584619138, −11.57754615700751342118738579815, −10.35686731877249381868935404615, −9.360211614288587877751584808687, −8.228096452384737037614173885936, −7.53300667685356143112092775831, −6.00467621767809857972921601858, −4.87752262758142254624901490751, −4.07701804646511460352838907064, −2.88015023871391165233455174295,
1.06568391578984764474728369134, 2.93648337889104611703500157957, 4.03617933401788104034199485399, 5.42103167874383518748810763811, 6.85088725814550800218979919673, 7.42585290291932953146723395861, 8.570832259285214721706850205714, 9.733781267601093933148486188033, 11.23479946185670083747295597405, 11.77797577196894023887109453869
