| L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.959 + 0.281i)4-s + (1.16 − 1.34i)5-s + (0.701 − 0.450i)7-s + (−0.415 − 0.909i)8-s + (1.49 + 0.962i)10-s + (0.558 − 3.88i)11-s + (3.85 + 2.47i)13-s + (0.546 + 0.630i)14-s + (0.841 − 0.540i)16-s + (1.11 + 0.328i)17-s + (3.59 − 1.05i)19-s + (−0.739 + 1.61i)20-s + 3.92·22-s + (−3.39 − 3.38i)23-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (−0.479 + 0.140i)4-s + (0.521 − 0.601i)5-s + (0.265 − 0.170i)7-s + (−0.146 − 0.321i)8-s + (0.473 + 0.304i)10-s + (0.168 − 1.17i)11-s + (1.06 + 0.687i)13-s + (0.145 + 0.168i)14-s + (0.210 − 0.135i)16-s + (0.271 + 0.0797i)17-s + (0.824 − 0.242i)19-s + (−0.165 + 0.361i)20-s + 0.836·22-s + (−0.708 − 0.705i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57737 + 0.256803i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57737 + 0.256803i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (3.39 + 3.38i)T \) |
| good | 5 | \( 1 + (-1.16 + 1.34i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.701 + 0.450i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.558 + 3.88i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-3.85 - 2.47i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.11 - 0.328i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-3.59 + 1.05i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.04 - 0.306i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.92 - 6.41i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (3.56 + 4.11i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (0.951 - 1.09i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.62 + 3.56i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 5.02T + 47T^{2} \) |
| 53 | \( 1 + (9.43 - 6.06i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-3.29 - 2.11i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (1.74 + 3.81i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (0.292 + 2.03i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.11 + 7.75i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (13.2 - 3.89i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (8.04 + 5.17i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (8.44 + 9.74i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (0.860 - 1.88i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-8.21 + 9.48i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.26824798224087396438136002842, −10.30738850685950310288690977974, −9.032371923456354662912952589928, −8.688242999808052697498602300396, −7.57512424230177339242823989553, −6.36681911957844410149772966498, −5.67272119966339618340254385890, −4.58673207101423653527091337354, −3.36202920434914397060752501392, −1.26955786495347353242970484816,
1.59174425575249070373236190032, 2.86548114675406274503756671702, 4.07103629201628729611691918102, 5.35610421534195677990468551250, 6.29368518031378414431740879503, 7.54491059472790015319755397478, 8.525181088535581447102170048357, 9.847420376445777229388348673099, 10.07441352664540716356457411425, 11.23989697227603974285659873611
