L(s) = 1 | + (0.173 + 0.984i)2-s + (0.504 − 2.86i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)5-s + 2.90·6-s + (0.773 + 0.649i)7-s + (−0.5 − 0.866i)8-s + (−5.12 − 1.86i)9-s + (−0.173 + 0.300i)10-s + (2.73 + 4.73i)11-s + (0.504 + 2.86i)12-s + (−5.97 + 2.17i)13-s + (−0.504 + 0.874i)14-s + (0.773 − 0.649i)15-s + (0.766 − 0.642i)16-s + (−0.490 − 0.178i)17-s + ⋯ |
L(s) = 1 | + (0.122 + 0.696i)2-s + (0.291 − 1.65i)3-s + (−0.469 + 0.171i)4-s + (0.118 + 0.0998i)5-s + 1.18·6-s + (0.292 + 0.245i)7-s + (−0.176 − 0.306i)8-s + (−1.70 − 0.621i)9-s + (−0.0549 + 0.0951i)10-s + (0.823 + 1.42i)11-s + (0.145 + 0.826i)12-s + (−1.65 + 0.603i)13-s + (−0.134 + 0.233i)14-s + (0.199 − 0.167i)15-s + (0.191 − 0.160i)16-s + (−0.119 − 0.0433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01343 - 0.118256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01343 - 0.118256i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 37 | \( 1 + (5.88 - 1.53i)T \) |
good | 3 | \( 1 + (-0.504 + 2.86i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.266 - 0.223i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.773 - 0.649i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-2.73 - 4.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.97 - 2.17i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.490 + 0.178i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.869 + 4.93i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (0.721 - 1.24i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.16 - 5.48i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 41 | \( 1 + (-1.03 + 0.376i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + (-5.71 + 9.90i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0706 + 0.0593i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-2.29 + 1.92i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (3.58 - 1.30i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.892 + 0.748i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0543 + 0.308i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + (-7.12 - 5.98i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (8.89 + 3.23i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (12.0 - 10.1i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.464 + 0.804i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.40724414514435799414318761010, −13.63895973436123503488484599952, −12.32421374721966689575954077475, −11.97861067913557097953870966237, −9.690104649648977455411997334394, −8.464910518162430431752206534220, −7.08390411798279786996538289576, −6.82882691075811576949023221098, −4.89447097243458177858122036153, −2.19179960255726958692821388548,
3.15439325335823151430107622770, 4.37859637584662036455254527585, 5.62298570593166065922861843854, 8.172007026784280048170172057084, 9.357933956818691750759675334508, 10.14640485824757525784815307111, 11.08979969309910324723610034393, 12.17004443038072295400497930740, 13.88408091760935021859654418875, 14.50524290523407492192357866859