| L(s) = 1 | + 5-s + 3·7-s − 13-s − 7·17-s − 5·19-s − 8·23-s + 25-s − 5·29-s + 10·31-s + 3·35-s + 7·37-s + 6·41-s + 10·43-s − 10·47-s + 2·49-s + 4·53-s + 4·59-s − 65-s + 6·67-s + 9·71-s + 2·73-s + 6·79-s + 7·83-s − 7·85-s − 3·91-s − 5·95-s − 14·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.277·13-s − 1.69·17-s − 1.14·19-s − 1.66·23-s + 1/5·25-s − 0.928·29-s + 1.79·31-s + 0.507·35-s + 1.15·37-s + 0.937·41-s + 1.52·43-s − 1.45·47-s + 2/7·49-s + 0.549·53-s + 0.520·59-s − 0.124·65-s + 0.733·67-s + 1.06·71-s + 0.234·73-s + 0.675·79-s + 0.768·83-s − 0.759·85-s − 0.314·91-s − 0.512·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.07854689260849, −13.71659316622823, −13.24553571198003, −12.63320362813543, −12.26964580193402, −11.48954071309933, −11.19825772091901, −10.81515909220734, −10.18417674137961, −9.616431541607559, −9.219805493375669, −8.463614878333041, −8.159183994412444, −7.750260470815406, −6.954427517803199, −6.354638237984945, −6.092900912793754, −5.307312048321161, −4.737939283881759, −4.195003960841689, −3.948789056450501, −2.643907209040748, −2.290023843994066, −1.843251727143892, −0.9354370444410662, 0,
0.9354370444410662, 1.843251727143892, 2.290023843994066, 2.643907209040748, 3.948789056450501, 4.195003960841689, 4.737939283881759, 5.307312048321161, 6.092900912793754, 6.354638237984945, 6.954427517803199, 7.750260470815406, 8.159183994412444, 8.463614878333041, 9.219805493375669, 9.616431541607559, 10.18417674137961, 10.81515909220734, 11.19825772091901, 11.48954071309933, 12.26964580193402, 12.63320362813543, 13.24553571198003, 13.71659316622823, 14.07854689260849
