| L(s) = 1 | − 5-s − 4·7-s − 11-s + 13-s − 5·17-s + 5·19-s + 6·23-s + 25-s + 4·29-s + 2·31-s + 4·35-s − 3·37-s − 5·41-s + 11·43-s + 3·47-s + 9·49-s + 12·53-s + 55-s + 14·59-s + 10·61-s − 65-s − 3·67-s − 6·71-s − 8·73-s + 4·77-s − 10·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.277·13-s − 1.21·17-s + 1.14·19-s + 1.25·23-s + 1/5·25-s + 0.742·29-s + 0.359·31-s + 0.676·35-s − 0.493·37-s − 0.780·41-s + 1.67·43-s + 0.437·47-s + 9/7·49-s + 1.64·53-s + 0.134·55-s + 1.82·59-s + 1.28·61-s − 0.124·65-s − 0.366·67-s − 0.712·71-s − 0.936·73-s + 0.455·77-s − 1.12·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.848258127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.848258127\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−13.45538786563417, −13.30994023024393, −12.82888141283047, −12.34556271880600, −11.58174643230976, −11.54644196250553, −10.68120487141786, −10.22104876033604, −9.931726133038742, −9.089612901386344, −8.875679566830927, −8.447030311204276, −7.482699978359673, −7.182796499387384, −6.753406922248271, −6.181406587249260, −5.603073424285235, −5.036265599464297, −4.350235485697567, −3.798213184220663, −3.195723923346571, −2.769460306262924, −2.141765363102468, −0.9481750932115709, −0.5313671506132881,
0.5313671506132881, 0.9481750932115709, 2.141765363102468, 2.769460306262924, 3.195723923346571, 3.798213184220663, 4.350235485697567, 5.036265599464297, 5.603073424285235, 6.181406587249260, 6.753406922248271, 7.182796499387384, 7.482699978359673, 8.447030311204276, 8.875679566830927, 9.089612901386344, 9.931726133038742, 10.22104876033604, 10.68120487141786, 11.54644196250553, 11.58174643230976, 12.34556271880600, 12.82888141283047, 13.30994023024393, 13.45538786563417
