| L(s) = 1 | + 468·11-s + 487·16-s + 1.28e3·31-s − 252·41-s − 2.36e4·61-s + 1.07e4·71-s + 1.28e4·81-s − 7.45e4·101-s + 7.83e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 2.27e5·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 3.86·11-s + 1.90·16-s + 1.34·31-s − 0.149·41-s − 6.35·61-s + 2.12·71-s + 1.96·81-s − 7.31·101-s + 5.34·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.34e−5·173-s + 7.35·176-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.973151606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.973151606\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 487 T^{4} + p^{16} T^{8} \) |
| 3 | $C_2^3$ | \( 1 - 1433 p^{2} T^{4} + p^{16} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 2302 p^{4} T^{4} + p^{16} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 117 T + p^{4} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 1054887458 T^{4} + p^{16} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 3503608657 T^{4} + p^{16} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 93383 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 122658570338 T^{4} + p^{16} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 45662 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 322 T + p^{4} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 2461256293058 T^{4} + p^{16} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 63 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 9927677992702 T^{4} + p^{16} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 39439575780578 T^{4} + p^{16} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 3858356729378 T^{4} + p^{16} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 20662622 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5908 T + p^{4} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 784493155081057 T^{4} + p^{16} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 2682 T + p^{4} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 912332767302863 T^{4} + p^{16} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 35389762 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 2296474800768143 T^{4} + p^{16} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 89664257 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 7754275002202178 T^{4} + p^{16} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.34993654278190739558830984912, −12.15734380651498095213666869302, −12.05162593439458748715641148122, −11.64657638048152489152536038077, −10.93355672274232673106367811075, −10.90744702740400900573203950121, −10.50225724794291976225766373480, −9.829607761320255637309591271315, −9.484098105064359227894850138010, −9.242155114789224846266109708936, −9.177913629982826713880892855750, −8.384638158077992173462140959204, −8.063860975641847468927630909873, −7.78840964758016182819100543359, −6.86588268561892082620820035512, −6.83686431525319678094815536014, −6.22615711411197775727415396872, −6.04710954053643655044123869346, −5.39003517703909450789188633394, −4.46400592614874281975720230262, −4.22350757572543351005885092455, −3.55177669062008212425601689380, −3.08830934884094570981025646812, −1.50550418686406373973513699175, −1.17890803720598540804519342286,
1.17890803720598540804519342286, 1.50550418686406373973513699175, 3.08830934884094570981025646812, 3.55177669062008212425601689380, 4.22350757572543351005885092455, 4.46400592614874281975720230262, 5.39003517703909450789188633394, 6.04710954053643655044123869346, 6.22615711411197775727415396872, 6.83686431525319678094815536014, 6.86588268561892082620820035512, 7.78840964758016182819100543359, 8.063860975641847468927630909873, 8.384638158077992173462140959204, 9.177913629982826713880892855750, 9.242155114789224846266109708936, 9.484098105064359227894850138010, 9.829607761320255637309591271315, 10.50225724794291976225766373480, 10.90744702740400900573203950121, 10.93355672274232673106367811075, 11.64657638048152489152536038077, 12.05162593439458748715641148122, 12.15734380651498095213666869302, 12.34993654278190739558830984912
