| L(s) = 1 | + 1.49·2-s − 3-s + 0.236·4-s − 1.30·5-s − 1.49·6-s − 2.63·8-s + 9-s − 1.94·10-s − 0.750·11-s − 0.236·12-s + 4.40·13-s + 1.30·15-s − 4.41·16-s + 4.00·17-s + 1.49·18-s − 5.38·19-s − 0.308·20-s − 1.12·22-s + 3.23·23-s + 2.63·24-s − 3.30·25-s + 6.58·26-s − 27-s − 3.07·29-s + 1.94·30-s + 9.02·31-s − 1.33·32-s + ⋯ |
| L(s) = 1 | + 1.05·2-s − 0.577·3-s + 0.118·4-s − 0.581·5-s − 0.610·6-s − 0.932·8-s + 0.333·9-s − 0.615·10-s − 0.226·11-s − 0.0683·12-s + 1.22·13-s + 0.335·15-s − 1.10·16-s + 0.970·17-s + 0.352·18-s − 1.23·19-s − 0.0688·20-s − 0.239·22-s + 0.674·23-s + 0.538·24-s − 0.661·25-s + 1.29·26-s − 0.192·27-s − 0.571·29-s + 0.355·30-s + 1.62·31-s − 0.235·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 11 | \( 1 + 0.750T + 11T^{2} \) |
| 13 | \( 1 - 4.40T + 13T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 3.07T + 29T^{2} \) |
| 31 | \( 1 - 9.02T + 31T^{2} \) |
| 37 | \( 1 - 0.222T + 37T^{2} \) |
| 43 | \( 1 - 0.320T + 43T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 - 1.91T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 0.199T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 9.59T + 83T^{2} \) |
| 89 | \( 1 + 4.12T + 89T^{2} \) |
| 97 | \( 1 - 6.19T + 97T^{2} \) |
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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
−7.71452664450490503126237467116, −6.66573049872356535920756557389, −6.14807099595505616645487719525, −5.54804182581830971588225239736, −4.75725690169997809561189996692, −4.10464204581321410703364213004, −3.53850668191420297671251803429, −2.65287422733582728844420404638, −1.25536569590324131901159666262, 0,
1.25536569590324131901159666262, 2.65287422733582728844420404638, 3.53850668191420297671251803429, 4.10464204581321410703364213004, 4.75725690169997809561189996692, 5.54804182581830971588225239736, 6.14807099595505616645487719525, 6.66573049872356535920756557389, 7.71452664450490503126237467116
