| L(s) = 1 | + 14·7-s + 114·9-s + 36·11-s − 1.47e3·23-s − 1.69e3·29-s − 4.77e3·37-s + 5.02e3·43-s − 2.20e3·49-s + 540·53-s + 1.59e3·63-s − 4.90e3·67-s − 6.30e3·71-s + 504·77-s − 7.96e3·79-s + 6.43e3·81-s + 4.10e3·99-s + 2.58e4·107-s − 1.40e4·109-s − 3.74e4·113-s − 2.83e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2/7·7-s + 1.40·9-s + 0.297·11-s − 2.79·23-s − 2.01·29-s − 3.48·37-s + 2.71·43-s − 0.918·49-s + 0.192·53-s + 0.402·63-s − 1.09·67-s − 1.24·71-s + 0.0850·77-s − 1.27·79-s + 0.980·81-s + 0.418·99-s + 2.26·107-s − 1.18·109-s − 2.93·113-s − 1.93·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8761383531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8761383531\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 p T + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 p T^{2} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 18 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 39794 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5758 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 252530 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 738 T + p^{4} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 846 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 492290 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2386 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1634 p^{2} T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2510 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 1859710 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 270 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 14384690 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14450830 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2450 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3150 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 56740994 T^{2} + p^{8} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3982 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 69825650 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 67615490 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18761474 T^{2} + p^{8} T^{4} \) |
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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
−10.27167720455696130526906975223, −9.627951553464494981825681262773, −9.142620243451089958744362312243, −9.040611583774664931166516435179, −8.135345277173917889542950938711, −8.004175492190739363139963866269, −7.38518943511079337723985823732, −7.10152686186682762229001942178, −6.69577614957683683232954837513, −5.83392955774283053011651104565, −5.80244210336991880678015217983, −5.15221363714083536214505316019, −4.33652625458475920103268608635, −4.15675712772178906034254704347, −3.72644517934444536430403603759, −3.06608341267545561159244261436, −1.94486211140506949202479301456, −1.91608973385211524446509398848, −1.28385552421691757621432388207, −0.21604643499578151548730473781,
0.21604643499578151548730473781, 1.28385552421691757621432388207, 1.91608973385211524446509398848, 1.94486211140506949202479301456, 3.06608341267545561159244261436, 3.72644517934444536430403603759, 4.15675712772178906034254704347, 4.33652625458475920103268608635, 5.15221363714083536214505316019, 5.80244210336991880678015217983, 5.83392955774283053011651104565, 6.69577614957683683232954837513, 7.10152686186682762229001942178, 7.38518943511079337723985823732, 8.004175492190739363139963866269, 8.135345277173917889542950938711, 9.040611583774664931166516435179, 9.142620243451089958744362312243, 9.627951553464494981825681262773, 10.27167720455696130526906975223
