L(s) = 1 | + 4·5-s − 6·13-s + 8·17-s + 11·25-s − 4·29-s − 2·37-s − 8·41-s − 7·49-s − 4·53-s − 10·61-s − 24·65-s + 6·73-s + 32·85-s + 16·89-s − 18·97-s − 20·101-s − 6·109-s + 16·113-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 1.66·13-s + 1.94·17-s + 11/5·25-s − 0.742·29-s − 0.328·37-s − 1.24·41-s − 49-s − 0.549·53-s − 1.28·61-s − 2.97·65-s + 0.702·73-s + 3.47·85-s + 1.69·89-s − 1.82·97-s − 1.99·101-s − 0.574·109-s + 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.626733000\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.626733000\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.01980574446063071553934149258, −10.55965367152125573686069092164, −9.816898356311968022813901953755, −9.372732192062624224981148085357, −7.909987862286350183375095910259, −6.81411763071537323600840753231, −5.66990658847027277525585177583, −5.01237677697104957198001932438, −3.02473421793484999957631922902, −1.74502909344688296171382532979,
1.74502909344688296171382532979, 3.02473421793484999957631922902, 5.01237677697104957198001932438, 5.66990658847027277525585177583, 6.81411763071537323600840753231, 7.909987862286350183375095910259, 9.372732192062624224981148085357, 9.816898356311968022813901953755, 10.55965367152125573686069092164, 12.01980574446063071553934149258