| L(s) = 1 | + 2·5-s + 7-s + 2·11-s − 6·13-s − 2·17-s + 19-s − 6·23-s − 25-s + 8·29-s − 8·31-s + 2·35-s − 10·37-s − 8·41-s − 8·43-s − 2·47-s + 49-s + 8·53-s + 4·55-s − 6·61-s − 12·65-s − 4·67-s + 6·73-s + 2·77-s − 14·83-s − 4·85-s + 16·89-s − 6·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s + 0.603·11-s − 1.66·13-s − 0.485·17-s + 0.229·19-s − 1.25·23-s − 1/5·25-s + 1.48·29-s − 1.43·31-s + 0.338·35-s − 1.64·37-s − 1.24·41-s − 1.21·43-s − 0.291·47-s + 1/7·49-s + 1.09·53-s + 0.539·55-s − 0.768·61-s − 1.48·65-s − 0.488·67-s + 0.702·73-s + 0.227·77-s − 1.53·83-s − 0.433·85-s + 1.69·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.943928178264536318543337756592, −7.05715572806805338765138973384, −6.59542281490148697971644358559, −5.61634618577600682996259224705, −5.06931442823840590739007603394, −4.28624816762728977806493992355, −3.26143005701318123664938461135, −2.17919827795219335758523745420, −1.66565006705413892970178336922, 0,
1.66565006705413892970178336922, 2.17919827795219335758523745420, 3.26143005701318123664938461135, 4.28624816762728977806493992355, 5.06931442823840590739007603394, 5.61634618577600682996259224705, 6.59542281490148697971644358559, 7.05715572806805338765138973384, 7.943928178264536318543337756592
