| L(s) = 1 | − 0.517·2-s + 3-s − 1.73·4-s − 5-s − 0.517·6-s + 1.87·7-s + 1.92·8-s + 9-s + 0.517·10-s − 0.451·11-s − 1.73·12-s + 0.973·13-s − 0.971·14-s − 15-s + 2.46·16-s − 0.380·17-s − 0.517·18-s − 0.189·19-s + 1.73·20-s + 1.87·21-s + 0.233·22-s − 4.33·23-s + 1.92·24-s + 25-s − 0.503·26-s + 27-s − 3.25·28-s + ⋯ |
| L(s) = 1 | − 0.365·2-s + 0.577·3-s − 0.866·4-s − 0.447·5-s − 0.211·6-s + 0.710·7-s + 0.682·8-s + 0.333·9-s + 0.163·10-s − 0.136·11-s − 0.500·12-s + 0.270·13-s − 0.259·14-s − 0.258·15-s + 0.616·16-s − 0.0923·17-s − 0.121·18-s − 0.0434·19-s + 0.387·20-s + 0.410·21-s + 0.0497·22-s − 0.904·23-s + 0.393·24-s + 0.200·25-s − 0.0987·26-s + 0.192·27-s − 0.615·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.517T + 2T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 + 0.451T + 11T^{2} \) |
| 13 | \( 1 - 0.973T + 13T^{2} \) |
| 17 | \( 1 + 0.380T + 17T^{2} \) |
| 19 | \( 1 + 0.189T + 19T^{2} \) |
| 23 | \( 1 + 4.33T + 23T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 + 5.05T + 37T^{2} \) |
| 41 | \( 1 + 0.126T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 - 5.41T + 47T^{2} \) |
| 53 | \( 1 + 8.20T + 53T^{2} \) |
| 59 | \( 1 + 4.35T + 59T^{2} \) |
| 61 | \( 1 + 1.68T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 2.27T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 4.58T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 8.26T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.86420901010898305497927454909, −7.41079511505829337941672211053, −6.39054533966332537447060930457, −5.36490565788835736026500909289, −4.77555392844787084630443168664, −3.96415131006066046625751472500, −3.44355221007343685478807048928, −2.17229995283676698205430689286, −1.29446598206464792968760858814, 0,
1.29446598206464792968760858814, 2.17229995283676698205430689286, 3.44355221007343685478807048928, 3.96415131006066046625751472500, 4.77555392844787084630443168664, 5.36490565788835736026500909289, 6.39054533966332537447060930457, 7.41079511505829337941672211053, 7.86420901010898305497927454909
