| L(s) = 1 | + 0.713·2-s − 1.49·4-s − 3.77·5-s + 1.42·7-s − 2.49·8-s − 2.69·10-s + 4.98·11-s + 2.98·13-s + 1.01·14-s + 1.20·16-s + 6.40·17-s − 2.91·19-s + 5.63·20-s + 3.55·22-s + 4·23-s + 9.26·25-s + 2.12·26-s − 2.12·28-s − 8.26·29-s + 4·31-s + 5.84·32-s + 4.57·34-s − 5.39·35-s − 2.50·37-s − 2.08·38-s + 9.40·40-s − 0.854·41-s + ⋯ |
| L(s) = 1 | + 0.504·2-s − 0.745·4-s − 1.68·5-s + 0.539·7-s − 0.880·8-s − 0.852·10-s + 1.50·11-s + 0.826·13-s + 0.272·14-s + 0.301·16-s + 1.55·17-s − 0.669·19-s + 1.25·20-s + 0.757·22-s + 0.834·23-s + 1.85·25-s + 0.417·26-s − 0.402·28-s − 1.53·29-s + 0.718·31-s + 1.03·32-s + 0.784·34-s − 0.911·35-s − 0.412·37-s − 0.337·38-s + 1.48·40-s − 0.133·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.323859939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323859939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 - 0.713T + 2T^{2} \) |
| 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 - 4.98T + 11T^{2} \) |
| 13 | \( 1 - 2.98T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 8.26T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2.50T + 37T^{2} \) |
| 41 | \( 1 + 0.854T + 41T^{2} \) |
| 43 | \( 1 - 7.20T + 43T^{2} \) |
| 47 | \( 1 - 4.40T + 47T^{2} \) |
| 53 | \( 1 + 3.96T + 53T^{2} \) |
| 59 | \( 1 - 5.55T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 0.981T + 67T^{2} \) |
| 73 | \( 1 - 5.36T + 73T^{2} \) |
| 79 | \( 1 - 0.350T + 79T^{2} \) |
| 83 | \( 1 - 8.06T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 6.57T + 97T^{2} \) |
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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
−10.89449309510614652453401633217, −9.526985738472985775008794407701, −8.679857435052978383422059609522, −8.074960814642490337499440913013, −7.11213222981206481197070802336, −5.92356118296599512684480884758, −4.78248823164934509074083745368, −3.88229310829785911376995900042, −3.47300824556356416896945052126, −0.987125445978223462941328083382,
0.987125445978223462941328083382, 3.47300824556356416896945052126, 3.88229310829785911376995900042, 4.78248823164934509074083745368, 5.92356118296599512684480884758, 7.11213222981206481197070802336, 8.074960814642490337499440913013, 8.679857435052978383422059609522, 9.526985738472985775008794407701, 10.89449309510614652453401633217
