| L(s) = 1 | − 1.65·2-s + 3-s + 0.729·4-s + 4.06·5-s − 1.65·6-s − 0.922·7-s + 2.09·8-s + 9-s − 6.71·10-s + 2.27·11-s + 0.729·12-s + 3.57·13-s + 1.52·14-s + 4.06·15-s − 4.92·16-s − 1.65·18-s + 1.72·19-s + 2.96·20-s − 0.922·21-s − 3.75·22-s − 5.09·23-s + 2.09·24-s + 11.5·25-s − 5.90·26-s + 27-s − 0.673·28-s + 2.20·29-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.577·3-s + 0.364·4-s + 1.81·5-s − 0.674·6-s − 0.348·7-s + 0.741·8-s + 0.333·9-s − 2.12·10-s + 0.684·11-s + 0.210·12-s + 0.991·13-s + 0.407·14-s + 1.04·15-s − 1.23·16-s − 0.389·18-s + 0.396·19-s + 0.663·20-s − 0.201·21-s − 0.799·22-s − 1.06·23-s + 0.428·24-s + 2.30·25-s − 1.15·26-s + 0.192·27-s − 0.127·28-s + 0.409·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455815007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455815007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 7 | \( 1 + 0.922T + 7T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 + 5.09T + 23T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 + 4.13T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 + 6.96T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 7.39T + 59T^{2} \) |
| 61 | \( 1 - 3.89T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 + 1.16T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 6.38T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−9.827168328241690214301753701579, −9.416656526364574795512348221971, −8.731350351012555279660006837323, −7.953610303924800944623557474873, −6.70582903165994512912104475001, −6.14344192733937856422884332598, −4.92374705666154757663658246091, −3.52126810644878989344123378558, −2.10167220170141520522193840579, −1.30043342296025974762632229268,
1.30043342296025974762632229268, 2.10167220170141520522193840579, 3.52126810644878989344123378558, 4.92374705666154757663658246091, 6.14344192733937856422884332598, 6.70582903165994512912104475001, 7.953610303924800944623557474873, 8.731350351012555279660006837323, 9.416656526364574795512348221971, 9.827168328241690214301753701579
