| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 2·12-s + 5·13-s + 16-s + 7·17-s − 18-s − 4·19-s + 23-s − 2·24-s − 5·26-s − 4·27-s − 4·31-s − 32-s − 7·34-s + 36-s − 3·37-s + 4·38-s + 10·39-s − 10·41-s − 4·43-s − 46-s + 9·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.38·13-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 0.917·19-s + 0.208·23-s − 0.408·24-s − 0.980·26-s − 0.769·27-s − 0.718·31-s − 0.176·32-s − 1.20·34-s + 1/6·36-s − 0.493·37-s + 0.648·38-s + 1.60·39-s − 1.56·41-s − 0.609·43-s − 0.147·46-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.66617359433688, −14.20164101393296, −13.65966993456115, −13.25843427069700, −12.64116959195606, −12.04253540204593, −11.56892113143134, −10.85984798288336, −10.48286820282577, −9.921942469858475, −9.379588161862498, −8.774544138899469, −8.508890206527499, −8.050553348342349, −7.514114019162188, −6.912170311263396, −6.288132425359521, −5.643631153852777, −5.159645604812672, −3.968181893640534, −3.681027115599620, −3.080950976792481, −2.439546393319590, −1.641283736052532, −1.163455692946330, 0,
1.163455692946330, 1.641283736052532, 2.439546393319590, 3.080950976792481, 3.681027115599620, 3.968181893640534, 5.159645604812672, 5.643631153852777, 6.288132425359521, 6.912170311263396, 7.514114019162188, 8.050553348342349, 8.508890206527499, 8.774544138899469, 9.379588161862498, 9.921942469858475, 10.48286820282577, 10.85984798288336, 11.56892113143134, 12.04253540204593, 12.64116959195606, 13.25843427069700, 13.65966993456115, 14.20164101393296, 14.66617359433688
