| L(s) = 1 | + (0.174 − 0.119i)2-s + (1.58 − 1.46i)3-s + (−0.714 + 1.82i)4-s + (−3.75 − 1.15i)5-s + (0.101 − 0.445i)6-s + (1.85 + 1.88i)7-s + (0.185 + 0.814i)8-s + (0.124 − 1.66i)9-s + (−0.792 + 0.244i)10-s + (−0.0750 − 1.00i)11-s + (1.54 + 3.93i)12-s + (−2.55 − 1.23i)13-s + (0.548 + 0.108i)14-s + (−7.63 + 3.67i)15-s + (−2.73 − 2.53i)16-s + (2.59 + 0.390i)17-s + ⋯ |
| L(s) = 1 | + (0.123 − 0.0841i)2-s + (0.914 − 0.848i)3-s + (−0.357 + 0.910i)4-s + (−1.67 − 0.517i)5-s + (0.0414 − 0.181i)6-s + (0.700 + 0.713i)7-s + (0.0657 + 0.288i)8-s + (0.0415 − 0.554i)9-s + (−0.250 + 0.0772i)10-s + (−0.0226 − 0.301i)11-s + (0.445 + 1.13i)12-s + (−0.708 − 0.341i)13-s + (0.146 + 0.0290i)14-s + (−1.97 + 0.949i)15-s + (−0.684 − 0.634i)16-s + (0.628 + 0.0947i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.876891 - 0.145079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.876891 - 0.145079i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.85 - 1.88i)T \) |
| good | 2 | \( 1 + (-0.174 + 0.119i)T + (0.730 - 1.86i)T^{2} \) |
| 3 | \( 1 + (-1.58 + 1.46i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (3.75 + 1.15i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (0.0750 + 1.00i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (2.55 + 1.23i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-2.59 - 0.390i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.64 + 2.84i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.67 - 0.403i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.95 + 3.70i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (1.33 + 2.31i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.02 - 2.60i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.845 - 3.70i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (0.320 - 1.40i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (4.46 - 3.04i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (0.112 - 0.286i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (7.68 - 2.37i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-2.16 - 5.52i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (5.38 + 9.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.08 - 6.37i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (10.5 + 7.22i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-2.63 + 4.56i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.55 - 0.748i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-1.00 + 13.4i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 7.32T + 97T^{2} \) |
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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
−15.46307332606959265373568836350, −14.38814002512399696180189594725, −13.11690441638696749867561062281, −12.19672677805260124855339839052, −11.55193306329613157713774811235, −8.921815850314833907264039589918, −7.971508095242935783885724481792, −7.63520363333484474756301733289, −4.66818634324634370193654695692, −3.02104773006964551102821617092,
3.70306290192136104625607400356, 4.70075074582980772801644727234, 7.22820443004217564205652731507, 8.374966613199968159570515761849, 9.849282580560856760532318188291, 10.78246188904373461321734290803, 12.08788323612837744862522952240, 14.20898127445435469940685244373, 14.52910133719226296766119991489, 15.39667041988770999616015350849
