| L(s) = 1 | − 4·7-s − 10·9-s + 20·17-s + 308·23-s − 25·25-s − 184·31-s − 796·41-s − 20·47-s − 674·49-s + 40·63-s − 1.85e3·71-s − 1.14e3·73-s + 128·79-s − 629·81-s + 1.74e3·89-s + 612·97-s − 924·103-s − 2.76e3·113-s − 80·119-s + 2.17e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 200·153-s + ⋯ |
| L(s) = 1 | − 0.215·7-s − 0.370·9-s + 0.285·17-s + 2.79·23-s − 1/5·25-s − 1.06·31-s − 3.03·41-s − 0.0620·47-s − 1.96·49-s + 0.0799·63-s − 3.10·71-s − 1.82·73-s + 0.182·79-s − 0.862·81-s + 2.08·89-s + 0.640·97-s − 0.883·103-s − 2.30·113-s − 0.0616·119-s + 1.63·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.105·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3221330803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3221330803\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 p^{2} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4294 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 1618 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 154 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 506 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100150 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 398 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 87190 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 40970 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 145758 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 402886 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 571942 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 928 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 570 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 64 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 397078 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 874 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 306 T + p^{3} T^{2} )^{2} \) |
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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
−9.823063564637589697592873910320, −8.979223811799974609828318438367, −8.610169038283606191480581355661, −8.607524411382800533073375916541, −7.75446558270328420543378685895, −7.46235520130240251951430703929, −7.04887711766449996305983638655, −6.63920719430993509275389339452, −6.28299725033784759464831980707, −5.64395239448636718819593636565, −5.30705935013748553616545547860, −4.80998197998308698453447678218, −4.57994682702098993088985655669, −3.68929252444468682623276406577, −3.18750979279518771343074296362, −3.11464703024210221721837428343, −2.33458386521237964221199457175, −1.47631416324560192179807151802, −1.27042552437866828832340469848, −0.13394954640635224560905830894,
0.13394954640635224560905830894, 1.27042552437866828832340469848, 1.47631416324560192179807151802, 2.33458386521237964221199457175, 3.11464703024210221721837428343, 3.18750979279518771343074296362, 3.68929252444468682623276406577, 4.57994682702098993088985655669, 4.80998197998308698453447678218, 5.30705935013748553616545547860, 5.64395239448636718819593636565, 6.28299725033784759464831980707, 6.63920719430993509275389339452, 7.04887711766449996305983638655, 7.46235520130240251951430703929, 7.75446558270328420543378685895, 8.607524411382800533073375916541, 8.610169038283606191480581355661, 8.979223811799974609828318438367, 9.823063564637589697592873910320
