| L(s) = 1 | − 3-s + 9-s + 4·13-s − 4·17-s + 4·23-s − 27-s + 2·29-s − 4·31-s − 6·37-s − 4·39-s + 2·41-s − 43-s − 7·49-s + 4·51-s − 10·53-s − 8·59-s − 8·61-s − 8·67-s − 4·69-s − 12·71-s + 10·73-s − 8·79-s + 81-s + 14·83-s − 2·87-s + 12·89-s + 4·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.10·13-s − 0.970·17-s + 0.834·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.640·39-s + 0.312·41-s − 0.152·43-s − 49-s + 0.560·51-s − 1.37·53-s − 1.04·59-s − 1.02·61-s − 0.977·67-s − 0.481·69-s − 1.42·71-s + 1.17·73-s − 0.900·79-s + 1/9·81-s + 1.53·83-s − 0.214·87-s + 1.27·89-s + 0.414·93-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
| \(L(1)\) |
\(\approx\) |
\(0.8418406188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8418406188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.13223004114774, −12.42714419768278, −12.18377194485038, −11.59314259219158, −10.95225723261492, −10.83055975835599, −10.53092265076294, −9.572554002500133, −9.355819596193032, −8.806810167232225, −8.318184210210791, −7.801273289105563, −7.204889980777543, −6.650057406913937, −6.369936014754466, −5.794013692664280, −5.282632262514847, −4.636323134745149, −4.367910520063467, −3.531459112786463, −3.187135529632702, −2.424559222089857, −1.569727157393638, −1.311324486351037, −0.2708223029389295,
0.2708223029389295, 1.311324486351037, 1.569727157393638, 2.424559222089857, 3.187135529632702, 3.531459112786463, 4.367910520063467, 4.636323134745149, 5.282632262514847, 5.794013692664280, 6.369936014754466, 6.650057406913937, 7.204889980777543, 7.801273289105563, 8.318184210210791, 8.806810167232225, 9.355819596193032, 9.572554002500133, 10.53092265076294, 10.83055975835599, 10.95225723261492, 11.59314259219158, 12.18377194485038, 12.42714419768278, 13.13223004114774
