L(s) = 1 | − 32·4-s + 592·9-s − 72·11-s + 768·16-s + 2.18e3·19-s + 9.74e3·29-s − 4.52e3·31-s − 1.89e4·36-s − 1.84e4·41-s + 2.30e3·44-s + 2.42e4·49-s − 1.80e5·59-s − 1.38e5·61-s − 1.63e4·64-s − 5.73e4·71-s − 6.98e4·76-s − 1.68e5·79-s + 1.77e5·81-s − 1.85e5·89-s − 4.26e4·99-s + 7.78e3·101-s − 4.82e3·109-s − 3.11e5·116-s − 1.15e5·121-s + 1.44e5·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 4-s + 2.43·9-s − 0.179·11-s + 3/4·16-s + 1.38·19-s + 2.15·29-s − 0.844·31-s − 2.43·36-s − 1.71·41-s + 0.179·44-s + 1.44·49-s − 6.76·59-s − 4.74·61-s − 1/2·64-s − 1.34·71-s − 1.38·76-s − 3.03·79-s + 2.99·81-s − 2.47·89-s − 0.437·99-s + 0.0759·101-s − 0.0388·109-s − 2.15·116-s − 0.716·121-s + 0.844·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.9357429105\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9357429105\) |
\(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
good | 3 | $D_4\times C_2$ | \( 1 - 592 T^{2} + 173458 T^{4} - 592 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3468 p T^{2} + 7278758 p^{2} T^{4} - 3468 p^{11} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 36 T + 59660 T^{2} + 36 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 86540 T^{2} - 3784960459802 T^{4} - 86540 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 1092 T + 5226780 T^{2} - 1092 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4296896 T^{2} + 73115643117858 T^{4} - 4296896 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 168 p T + 24870890 T^{2} - 168 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2260 T - 4802604 T^{2} + 2260 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 81075300 T^{2} + 5174214699551798 T^{4} - 81075300 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 9220 T + 191811102 T^{2} + 9220 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 307121776 T^{2} + 66493657362749938 T^{4} - 307121776 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 818687220 T^{2} + 271437169988573798 T^{4} - 818687220 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1603427228 T^{2} + 992023260615384310 T^{4} - 1603427228 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 90404 T + 3449372988 T^{2} + 90404 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 69000 T + 2784511178 T^{2} + 69000 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 3438652308 T^{2} + 5735146176860862518 T^{4} - 3438652308 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 28668 T + 3332675492 T^{2} + 28668 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 6138404788 T^{2} + 17643912066464393670 T^{4} - 6138404788 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 84064 T + 4137892006 T^{2} + 84064 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4727788228 T^{2} + 35719446237480418710 T^{4} - 4727788228 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 92620 T + 13143968022 T^{2} + 92620 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 28945542972 T^{2} + \)\(34\!\cdots\!10\)\( T^{4} - 28945542972 p^{10} T^{6} + p^{20} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.98757781218594973133115870854, −6.47624647882894459242544956580, −6.32225895189913788447326419851, −6.13172444207821644923400486541, −5.77644640849140991196864230450, −5.69599434135675158464679069133, −5.27502655931792704818662700044, −4.91076561495714424923936568113, −4.68940879937535060625406298549, −4.66197663612106823366654314072, −4.31966823979743706088062782619, −4.30542046913951442647374910370, −4.02481829338212262406214068930, −3.32659167346692373838370124474, −3.27596891446237640375953424171, −3.17746977429690549705256963460, −2.85084614691639572370735438477, −2.47354201959764268132148851212, −1.93858970345071507794288773128, −1.45490348744143616148447274799, −1.33035435032325176047832885419, −1.28978000147736598621030073539, −1.22022309103391759501140567243, −0.27551089067402655469312379277, −0.17792521344228198541589631241,
0.17792521344228198541589631241, 0.27551089067402655469312379277, 1.22022309103391759501140567243, 1.28978000147736598621030073539, 1.33035435032325176047832885419, 1.45490348744143616148447274799, 1.93858970345071507794288773128, 2.47354201959764268132148851212, 2.85084614691639572370735438477, 3.17746977429690549705256963460, 3.27596891446237640375953424171, 3.32659167346692373838370124474, 4.02481829338212262406214068930, 4.30542046913951442647374910370, 4.31966823979743706088062782619, 4.66197663612106823366654314072, 4.68940879937535060625406298549, 4.91076561495714424923936568113, 5.27502655931792704818662700044, 5.69599434135675158464679069133, 5.77644640849140991196864230450, 6.13172444207821644923400486541, 6.32225895189913788447326419851, 6.47624647882894459242544956580, 6.98757781218594973133115870854