| L(s) = 1 | − 500·5-s + 7.98e3·11-s + 5.06e4·19-s + 1.71e5·25-s − 3.05e5·29-s − 2.47e5·31-s − 7.93e5·41-s + 1.12e6·49-s − 3.99e6·55-s − 6.04e5·59-s − 5.66e6·61-s + 2.01e6·71-s − 1.50e7·79-s − 1.53e7·89-s − 2.53e7·95-s − 2.25e7·101-s − 2.34e7·109-s + 8.88e6·121-s − 4.68e7·125-s + 127-s + 131-s + 137-s + 139-s + 1.52e8·145-s + 149-s + 151-s + 1.23e8·155-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.80·11-s + 1.69·19-s + 11/5·25-s − 2.32·29-s − 1.49·31-s − 1.79·41-s + 1.36·49-s − 3.23·55-s − 0.383·59-s − 3.19·61-s + 0.668·71-s − 3.43·79-s − 2.30·89-s − 3.02·95-s − 2.18·101-s − 1.73·109-s + 0.455·121-s − 2.14·125-s + 4.15·145-s + 2.66·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1504231193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1504231193\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 p^{3} T + p^{7} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 1125802 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3994 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 116316134 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 397058622 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 25320 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2367161790 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 152664 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 123776 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 75696805270 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 396530 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 347519328310 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 984199174302 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 813245470198 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 302354 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2830198 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1875692967338 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1007580 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16312522745698 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 7517832 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26186045040870 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 7650250 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 60474559225090 T^{2} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.86171847026634351430671352350, −11.08971006696745304097515581547, −11.02980655422956737428135054231, −10.07511645293896848159055621527, −9.482514805404702157579857426051, −8.876677708870349062858585918377, −8.866601705884630720654249394431, −7.74516178536508037646133817443, −7.64166925900676663574935570567, −7.02682034204539483424304163170, −6.67024807192585293947618394415, −5.64411174901586297619185261546, −5.30597810307436906077004231667, −4.23238173185711688885304875640, −4.04087301358319925743917687115, −3.43075825681166467753890605453, −2.95648999846202006615127155673, −1.51070596144674787115739690215, −1.32697229504227886533526677975, −0.10616576183936039173569354817,
0.10616576183936039173569354817, 1.32697229504227886533526677975, 1.51070596144674787115739690215, 2.95648999846202006615127155673, 3.43075825681166467753890605453, 4.04087301358319925743917687115, 4.23238173185711688885304875640, 5.30597810307436906077004231667, 5.64411174901586297619185261546, 6.67024807192585293947618394415, 7.02682034204539483424304163170, 7.64166925900676663574935570567, 7.74516178536508037646133817443, 8.866601705884630720654249394431, 8.876677708870349062858585918377, 9.482514805404702157579857426051, 10.07511645293896848159055621527, 11.02980655422956737428135054231, 11.08971006696745304097515581547, 11.86171847026634351430671352350
