| L(s) = 1 | − 16·4-s − 34·9-s − 92·11-s + 256·16-s − 868·19-s + 544·36-s − 2.49e3·41-s + 1.47e3·44-s − 4.80e3·49-s + 476·59-s − 4.09e3·64-s + 1.38e4·76-s − 5.40e3·81-s − 1.09e4·89-s + 3.12e3·99-s − 2.29e4·121-s + 127-s + 131-s + 137-s + 139-s − 8.70e3·144-s + 149-s + 151-s + 157-s + 163-s + 3.98e4·164-s + 167-s + ⋯ |
| L(s) = 1 | − 4-s − 0.419·9-s − 0.760·11-s + 16-s − 2.40·19-s + 0.419·36-s − 1.48·41-s + 0.760·44-s − 2·49-s + 0.136·59-s − 64-s + 2.40·76-s − 0.823·81-s − 1.38·89-s + 0.319·99-s − 1.56·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 0.419·144-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 1.48·164-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.01498943431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01498943431\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 34 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 46 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 162434 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 434 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 1246 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 5426402 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 238 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13944286 T^{2} + p^{8} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 33567554 T^{2} + p^{8} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30209954 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5474 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 77418238 T^{2} + p^{8} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.11963295755118933004549876272, −11.50552217366292155303976898647, −11.02352653389552655235856276628, −10.41187268765315087249785992385, −10.18192317420214882745402545868, −9.585178715396212186268281386429, −8.879847123876168789511497769686, −8.606346223542007273601008152053, −8.096727312345969306075980035451, −7.72265187498301928274875181130, −6.72362175252496037700361006432, −6.39130652477099410235905907291, −5.61648759722625936015978526979, −5.09339818461974167136794760336, −4.51397276797590560348913566779, −3.93619976125295992971919352838, −3.16584064642125537301787035387, −2.36376083442771196809807971657, −1.44762917638542753739002023695, −0.04411950339036027512986425081,
0.04411950339036027512986425081, 1.44762917638542753739002023695, 2.36376083442771196809807971657, 3.16584064642125537301787035387, 3.93619976125295992971919352838, 4.51397276797590560348913566779, 5.09339818461974167136794760336, 5.61648759722625936015978526979, 6.39130652477099410235905907291, 6.72362175252496037700361006432, 7.72265187498301928274875181130, 8.096727312345969306075980035451, 8.606346223542007273601008152053, 8.879847123876168789511497769686, 9.585178715396212186268281386429, 10.18192317420214882745402545868, 10.41187268765315087249785992385, 11.02352653389552655235856276628, 11.50552217366292155303976898647, 12.11963295755118933004549876272
