| L(s) = 1 | − 2·4-s + 7-s + 4·16-s − 8·19-s − 5·25-s − 2·28-s + 7·31-s + 10·37-s − 13·43-s − 6·49-s − 13·61-s − 8·64-s − 11·67-s − 17·73-s + 16·76-s − 13·79-s − 5·97-s + 10·100-s − 13·103-s + 19·109-s + 4·112-s + ⋯ |
| L(s) = 1 | − 4-s + 0.377·7-s + 16-s − 1.83·19-s − 25-s − 0.377·28-s + 1.25·31-s + 1.64·37-s − 1.98·43-s − 6/7·49-s − 1.66·61-s − 64-s − 1.34·67-s − 1.98·73-s + 1.83·76-s − 1.46·79-s − 0.507·97-s + 100-s − 1.28·103-s + 1.81·109-s + 0.377·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.976353361441638594685535465892, −8.312821868280382142067857193844, −7.74586259697582074911205944081, −6.48490940707721769963949397838, −5.77793286940956698342983625285, −4.61144621747839189405000263785, −4.25243634621054323274097662365, −2.98313723349999362339939315781, −1.60868123054944801912731821594, 0,
1.60868123054944801912731821594, 2.98313723349999362339939315781, 4.25243634621054323274097662365, 4.61144621747839189405000263785, 5.77793286940956698342983625285, 6.48490940707721769963949397838, 7.74586259697582074911205944081, 8.312821868280382142067857193844, 8.976353361441638594685535465892
