| L(s) = 1 | + 13-s + 2·17-s + 4·19-s − 4·23-s + 6·29-s − 8·31-s + 6·37-s + 2·41-s + 4·43-s − 7·49-s + 6·53-s − 2·61-s + 8·67-s − 6·73-s + 4·79-s + 12·83-s − 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.277·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1.11·29-s − 1.43·31-s + 0.986·37-s + 0.312·41-s + 0.609·43-s − 49-s + 0.824·53-s − 0.256·61-s + 0.977·67-s − 0.702·73-s + 0.450·79-s + 1.31·83-s − 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.606474795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.606474795\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.04863061772638, −13.14392323190448, −13.03232671381844, −12.20810564608649, −11.98540328553618, −11.33079195136814, −10.94583330706029, −10.29464286096973, −9.893097216142637, −9.350316295082681, −8.908515711169095, −8.233515385571939, −7.770195497373093, −7.369876334220086, −6.687631793947119, −6.122088039300490, −5.653538761671218, −5.092029795747194, −4.478983778983331, −3.819427427191389, −3.342855997207120, −2.656185397731873, −1.988744385720954, −1.218005795907337, −0.5524808123357553,
0.5524808123357553, 1.218005795907337, 1.988744385720954, 2.656185397731873, 3.342855997207120, 3.819427427191389, 4.478983778983331, 5.092029795747194, 5.653538761671218, 6.122088039300490, 6.687631793947119, 7.369876334220086, 7.770195497373093, 8.233515385571939, 8.908515711169095, 9.350316295082681, 9.893097216142637, 10.29464286096973, 10.94583330706029, 11.33079195136814, 11.98540328553618, 12.20810564608649, 13.03232671381844, 13.14392323190448, 14.04863061772638
