| L(s) = 1 | + 3-s + 9-s − 2·11-s − 2·13-s + 2·17-s + 4·19-s + 6·23-s + 27-s − 4·31-s − 2·33-s + 2·37-s − 2·39-s + 6·41-s + 43-s − 2·47-s − 7·49-s + 2·51-s − 14·53-s + 4·57-s + 14·59-s + 14·61-s + 12·67-s + 6·69-s − 6·73-s − 4·79-s + 81-s + 14·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.25·23-s + 0.192·27-s − 0.718·31-s − 0.348·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.152·43-s − 0.291·47-s − 49-s + 0.280·51-s − 1.92·53-s + 0.529·57-s + 1.82·59-s + 1.79·61-s + 1.46·67-s + 0.722·69-s − 0.702·73-s − 0.450·79-s + 1/9·81-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.361967384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.361967384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.11152801872777, −12.70223866747415, −12.16925192240445, −11.62359102004978, −11.05710166377178, −10.82691647699535, −10.02413208129020, −9.703170021327163, −9.369736482288628, −8.806157353268375, −8.178271250963239, −7.828178353862181, −7.414526257533384, −6.835074795778759, −6.466560610495665, −5.493985357311646, −5.320655107070353, −4.794027571242645, −4.091243734148451, −3.519892303495886, −2.991279406315901, −2.559609034787945, −1.903055372234288, −1.160045466203556, −0.5266607313012952,
0.5266607313012952, 1.160045466203556, 1.903055372234288, 2.559609034787945, 2.991279406315901, 3.519892303495886, 4.091243734148451, 4.794027571242645, 5.320655107070353, 5.493985357311646, 6.466560610495665, 6.835074795778759, 7.414526257533384, 7.828178353862181, 8.178271250963239, 8.806157353268375, 9.369736482288628, 9.703170021327163, 10.02413208129020, 10.82691647699535, 11.05710166377178, 11.62359102004978, 12.16925192240445, 12.70223866747415, 13.11152801872777
