L(s) = 1 | + 3-s − 5-s + 9-s + 2·11-s − 2·13-s − 15-s + 6·17-s − 6·19-s − 8·23-s + 25-s + 27-s + 2·29-s + 2·33-s − 37-s − 2·39-s + 2·41-s − 45-s − 6·47-s − 7·49-s + 6·51-s + 2·53-s − 2·55-s − 6·57-s − 4·61-s + 2·65-s + 4·67-s − 8·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 1.37·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.348·33-s − 0.164·37-s − 0.320·39-s + 0.312·41-s − 0.149·45-s − 0.875·47-s − 49-s + 0.840·51-s + 0.274·53-s − 0.269·55-s − 0.794·57-s − 0.512·61-s + 0.248·65-s + 0.488·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.59253834998304981127829363554, −6.73930855718630220665536926732, −6.16438458911315798529213715846, −5.29300576592538914904318356627, −4.39270416355988857094443033533, −3.87839445505629607036708730213, −3.12059952267117642062641079318, −2.22333631027859198478866035350, −1.35479777565574537817999109382, 0,
1.35479777565574537817999109382, 2.22333631027859198478866035350, 3.12059952267117642062641079318, 3.87839445505629607036708730213, 4.39270416355988857094443033533, 5.29300576592538914904318356627, 6.16438458911315798529213715846, 6.73930855718630220665536926732, 7.59253834998304981127829363554