| L(s) = 1 | + (−0.669 + 0.743i)2-s + (−0.338 + 3.21i)3-s + (−0.104 − 0.994i)4-s + (3.76 − 0.801i)5-s + (−2.16 − 2.40i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (−7.30 − 1.55i)9-s + (−1.92 + 3.33i)10-s + (−0.809 + 3.21i)11-s + 3.23·12-s + (2.79 + 0.593i)13-s + (0.104 − 0.994i)14-s + (1.30 + 12.4i)15-s + (−0.978 + 0.207i)16-s + (−3.53 + 0.752i)17-s + ⋯ |
| L(s) = 1 | + (−0.473 + 0.525i)2-s + (−0.195 + 1.85i)3-s + (−0.0522 − 0.497i)4-s + (1.68 − 0.358i)5-s + (−0.884 − 0.981i)6-s + (−0.305 + 0.222i)7-s + (0.286 + 0.207i)8-s + (−2.43 − 0.517i)9-s + (−0.609 + 1.05i)10-s + (−0.243 + 0.969i)11-s + 0.934·12-s + (0.774 + 0.164i)13-s + (0.0279 − 0.265i)14-s + (0.336 + 3.20i)15-s + (−0.244 + 0.0519i)16-s + (−0.858 + 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.198209 + 1.14607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.198209 + 1.14607i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 11 | \( 1 + (0.809 - 3.21i)T \) |
| 19 | \( 1 + (0.658 - 4.30i)T \) |
| good | 3 | \( 1 + (0.338 - 3.21i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-3.76 + 0.801i)T + (4.56 - 2.03i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-2.79 - 0.593i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (3.53 - 0.752i)T + (15.5 - 6.91i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.104 - 0.994i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.0278 + 0.0857i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.92 + 2.12i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.442 + 4.21i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (5.73 + 9.93i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.913 - 0.406i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-6.61 - 1.40i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-12.7 - 5.67i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (2.77 + 3.08i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-6.66 + 11.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-15.2 + 3.24i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-6.34 - 2.82i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (9.68 - 10.7i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.54 + 4.75i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (2.97 - 5.14i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.86 + 6.51i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−11.05299411541485183773909947602, −10.29078879590031580204153397580, −9.782810088752768271242215434457, −9.150351016172875723992254268912, −8.467499273917711934577845551602, −6.57683328955881931293980880093, −5.72670721834213760912634502472, −5.10175083788597964386311803326, −3.95056010795132566588940658811, −2.17645627680524211755005426208,
0.909259032944458430390312007455, 2.14213796943196138784393067481, 2.95040988090560831260210854722, 5.43247440661493273955575098129, 6.44304890972768484792710669547, 6.76183324892172006093834348191, 8.157462807679212510143272416535, 8.857456723053196679746925554562, 9.946093764147112679069644572133, 11.05165936159760044088671569267
