| L(s) = 1 | + 0.603·2-s + 2.40·3-s − 1.63·4-s + 0.408·5-s + 1.44·6-s + 7-s − 2.19·8-s + 2.76·9-s + 0.246·10-s + 11-s − 3.92·12-s + 4.04·13-s + 0.603·14-s + 0.980·15-s + 1.94·16-s + 1.62·17-s + 1.67·18-s + 1.34·19-s − 0.667·20-s + 2.40·21-s + 0.603·22-s − 2.61·23-s − 5.27·24-s − 4.83·25-s + 2.44·26-s − 0.552·27-s − 1.63·28-s + ⋯ |
| L(s) = 1 | + 0.426·2-s + 1.38·3-s − 0.817·4-s + 0.182·5-s + 0.591·6-s + 0.377·7-s − 0.775·8-s + 0.923·9-s + 0.0779·10-s + 0.301·11-s − 1.13·12-s + 1.12·13-s + 0.161·14-s + 0.253·15-s + 0.486·16-s + 0.392·17-s + 0.393·18-s + 0.307·19-s − 0.149·20-s + 0.524·21-s + 0.128·22-s − 0.545·23-s − 1.07·24-s − 0.966·25-s + 0.478·26-s − 0.106·27-s − 0.309·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.578578293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.578578293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 0.603T + 2T^{2} \) |
| 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 - 0.408T + 5T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 - 1.34T + 19T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 2.31T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 - 7.96T + 41T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 6.60T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 + 1.58T + 97T^{2} \) |
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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
−8.752096477721342223469987307702, −7.988298007297953011760922032149, −7.49144356205779568950867985911, −6.13707032693844820122383500772, −5.65901930107490310552814692196, −4.46569351671193663296474826004, −3.90310252651499306984511420915, −3.21909880760208827132387462125, −2.25653100841345194262413343375, −1.06826151581588633746258099178,
1.06826151581588633746258099178, 2.25653100841345194262413343375, 3.21909880760208827132387462125, 3.90310252651499306984511420915, 4.46569351671193663296474826004, 5.65901930107490310552814692196, 6.13707032693844820122383500772, 7.49144356205779568950867985911, 7.988298007297953011760922032149, 8.752096477721342223469987307702
