| L(s) = 1 | − 5·5-s + 34·7-s + 18·11-s + 12·13-s − 106·17-s − 44·19-s + 56·23-s + 25·25-s + 270·29-s + 204·31-s − 170·35-s + 120·37-s + 80·41-s + 536·43-s − 536·47-s + 813·49-s + 542·53-s − 90·55-s − 174·59-s + 186·61-s − 60·65-s + 332·67-s − 132·71-s − 602·73-s + 612·77-s − 548·79-s − 492·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.83·7-s + 0.493·11-s + 0.256·13-s − 1.51·17-s − 0.531·19-s + 0.507·23-s + 1/5·25-s + 1.72·29-s + 1.18·31-s − 0.821·35-s + 0.533·37-s + 0.304·41-s + 1.90·43-s − 1.66·47-s + 2.37·49-s + 1.40·53-s − 0.220·55-s − 0.383·59-s + 0.390·61-s − 0.114·65-s + 0.605·67-s − 0.220·71-s − 0.965·73-s + 0.905·77-s − 0.780·79-s − 0.650·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.189622919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189622919\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| good | 7 | \( 1 - 34 T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 106 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 270 T + p^{3} T^{2} \) |
| 31 | \( 1 - 204 T + p^{3} T^{2} \) |
| 37 | \( 1 - 120 T + p^{3} T^{2} \) |
| 41 | \( 1 - 80 T + p^{3} T^{2} \) |
| 43 | \( 1 - 536 T + p^{3} T^{2} \) |
| 47 | \( 1 + 536 T + p^{3} T^{2} \) |
| 53 | \( 1 - 542 T + p^{3} T^{2} \) |
| 59 | \( 1 + 174 T + p^{3} T^{2} \) |
| 61 | \( 1 - 186 T + p^{3} T^{2} \) |
| 67 | \( 1 - 332 T + p^{3} T^{2} \) |
| 71 | \( 1 + 132 T + p^{3} T^{2} \) |
| 73 | \( 1 + 602 T + p^{3} T^{2} \) |
| 79 | \( 1 + 548 T + p^{3} T^{2} \) |
| 83 | \( 1 + 492 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1052 T + p^{3} T^{2} \) |
| 97 | \( 1 - 482 T + p^{3} T^{2} \) |
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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
−11.25808531216201273573588106224, −10.31275244905125507314447820928, −8.790479079649325559287251080930, −8.387197820463201716795574761840, −7.32071529884128773136050174955, −6.24459586299124960364520383679, −4.76675895001472495067985552673, −4.26902889512726914666448585289, −2.43922807156252713002282077449, −1.06958996659169565273810351547,
1.06958996659169565273810351547, 2.43922807156252713002282077449, 4.26902889512726914666448585289, 4.76675895001472495067985552673, 6.24459586299124960364520383679, 7.32071529884128773136050174955, 8.387197820463201716795574761840, 8.790479079649325559287251080930, 10.31275244905125507314447820928, 11.25808531216201273573588106224
