| L(s) = 1 | + 3-s + 4·7-s + 9-s + 2·11-s − 4·13-s − 17-s − 4·19-s + 4·21-s + 8·23-s + 27-s + 8·29-s + 2·33-s + 2·37-s − 4·39-s − 4·41-s − 6·43-s + 12·47-s + 9·49-s − 51-s − 14·53-s − 4·57-s + 2·61-s + 4·63-s − 2·67-s + 8·69-s + 14·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.242·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s + 0.192·27-s + 1.48·29-s + 0.348·33-s + 0.328·37-s − 0.640·39-s − 0.624·41-s − 0.914·43-s + 1.75·47-s + 9/7·49-s − 0.140·51-s − 1.92·53-s − 0.529·57-s + 0.256·61-s + 0.503·63-s − 0.244·67-s + 0.963·69-s + 1.66·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.420363382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.420363382\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.90773586084484, −15.89400087910077, −15.23896861324889, −14.78457964884341, −14.49677781559359, −13.85318663626678, −13.29938627026804, −12.40651212742355, −12.11958326832165, −11.28389430014044, −10.83957711876483, −10.20304473756546, −9.365553675676884, −8.889150778998765, −8.223294600707479, −7.806548995716002, −6.955022154746035, −6.546366606602164, −5.372558488714197, −4.710195530722102, −4.422895717055419, −3.332062423090483, −2.480880448754583, −1.808138584010318, −0.8814587100583054,
0.8814587100583054, 1.808138584010318, 2.480880448754583, 3.332062423090483, 4.422895717055419, 4.710195530722102, 5.372558488714197, 6.546366606602164, 6.955022154746035, 7.806548995716002, 8.223294600707479, 8.889150778998765, 9.365553675676884, 10.20304473756546, 10.83957711876483, 11.28389430014044, 12.11958326832165, 12.40651212742355, 13.29938627026804, 13.85318663626678, 14.49677781559359, 14.78457964884341, 15.23896861324889, 15.89400087910077, 16.90773586084484
