| L(s) = 1 | − 16·4-s − 43·9-s + 1.11e3·11-s + 256·16-s + 4.07e3·19-s + 1.00e4·29-s + 1.13e4·31-s + 688·36-s − 4.84e3·41-s − 1.77e4·44-s − 2.40e3·49-s − 1.14e4·59-s − 7.22e4·61-s − 4.09e3·64-s + 3.21e4·71-s − 6.52e4·76-s + 1.28e5·79-s − 5.72e4·81-s + 1.43e5·89-s − 4.77e4·99-s − 1.14e5·101-s − 1.76e5·109-s − 1.60e5·116-s + 6.01e5·121-s − 1.82e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.176·9-s + 2.76·11-s + 1/4·16-s + 2.59·19-s + 2.20·29-s + 2.12·31-s + 0.0884·36-s − 0.450·41-s − 1.38·44-s − 1/7·49-s − 0.428·59-s − 2.48·61-s − 1/8·64-s + 0.757·71-s − 1.29·76-s + 2.31·79-s − 0.968·81-s + 1.91·89-s − 0.489·99-s − 1.11·101-s − 1.42·109-s − 1.10·116-s + 3.73·121-s − 1.06·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.887761490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.887761490\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 43 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 555 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 684505 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 616633 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2038 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11359786 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5001 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5696 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 107305510 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2424 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158818 p^{2} T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 77834555 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 196503370 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5724 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 592 p T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1669488602 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16080 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2331209138 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 64147 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 3418207370 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 71676 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 5633870111 T^{2} + p^{10} T^{4} \) |
| 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
−10.79436551992783854040129524485, −10.46105753580863299980992283273, −9.665584866555970450980743718289, −9.456745655161154827255724815166, −9.301828437467604673834132144747, −8.531504222638364019542957352750, −8.221087832401596413136055544246, −7.63559629086931441032039897191, −6.96674019568112547110125440797, −6.49643343210158103303950850906, −6.25026670490442218478676980501, −5.53427091592031866883421927762, −4.66975685954181328755070924381, −4.63320630788029861859570397367, −3.75633852047851529771488045172, −3.28007492211831578059512920654, −2.77116584728048343214059821393, −1.55533415445948253457156174616, −1.06191224531082288781426379242, −0.73413857434771612288531027909,
0.73413857434771612288531027909, 1.06191224531082288781426379242, 1.55533415445948253457156174616, 2.77116584728048343214059821393, 3.28007492211831578059512920654, 3.75633852047851529771488045172, 4.63320630788029861859570397367, 4.66975685954181328755070924381, 5.53427091592031866883421927762, 6.25026670490442218478676980501, 6.49643343210158103303950850906, 6.96674019568112547110125440797, 7.63559629086931441032039897191, 8.221087832401596413136055544246, 8.531504222638364019542957352750, 9.301828437467604673834132144747, 9.456745655161154827255724815166, 9.665584866555970450980743718289, 10.46105753580863299980992283273, 10.79436551992783854040129524485
