| L(s) = 1 | − 1.44·2-s + 0.801·3-s + 0.0881·4-s + 5-s − 1.15·6-s + 7-s + 2.76·8-s − 2.35·9-s − 1.44·10-s − 5.49·11-s + 0.0706·12-s − 6.04·13-s − 1.44·14-s + 0.801·15-s − 4.16·16-s − 17-s + 3.40·18-s + 5.98·19-s + 0.0881·20-s + 0.801·21-s + 7.93·22-s + 0.780·23-s + 2.21·24-s + 25-s + 8.74·26-s − 4.29·27-s + 0.0881·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.462·3-s + 0.0440·4-s + 0.447·5-s − 0.473·6-s + 0.377·7-s + 0.976·8-s − 0.785·9-s − 0.456·10-s − 1.65·11-s + 0.0204·12-s − 1.67·13-s − 0.386·14-s + 0.207·15-s − 1.04·16-s − 0.242·17-s + 0.802·18-s + 1.37·19-s + 0.0197·20-s + 0.174·21-s + 1.69·22-s + 0.162·23-s + 0.452·24-s + 0.200·25-s + 1.71·26-s − 0.826·27-s + 0.0166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 1.44T + 2T^{2} \) |
| 3 | \( 1 - 0.801T + 3T^{2} \) |
| 11 | \( 1 + 5.49T + 11T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 19 | \( 1 - 5.98T + 19T^{2} \) |
| 23 | \( 1 - 0.780T + 23T^{2} \) |
| 29 | \( 1 + 0.841T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 6.67T + 37T^{2} \) |
| 41 | \( 1 + 0.841T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 5.86T + 53T^{2} \) |
| 59 | \( 1 - 6.29T + 59T^{2} \) |
| 61 | \( 1 - 4.51T + 61T^{2} \) |
| 67 | \( 1 - 7.50T + 67T^{2} \) |
| 71 | \( 1 + 8.96T + 71T^{2} \) |
| 73 | \( 1 + 7.64T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 - 2.72T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−9.829195198907484900237764029841, −9.573302981833051921198019030872, −8.416332080656630596604865992901, −7.84960300483568627773503133814, −7.13375973109552762197150930748, −5.38823114395619834616469261954, −4.90887589924337165947098772892, −3.03103193932449988291144459693, −2.01235534209284527902049367346, 0,
2.01235534209284527902049367346, 3.03103193932449988291144459693, 4.90887589924337165947098772892, 5.38823114395619834616469261954, 7.13375973109552762197150930748, 7.84960300483568627773503133814, 8.416332080656630596604865992901, 9.573302981833051921198019030872, 9.829195198907484900237764029841
