| L(s) = 1 | + 12·9-s + 8·11-s − 12·25-s − 40·43-s − 12·49-s − 24·67-s + 64·81-s + 96·99-s + 56·107-s + 72·113-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 92·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 4·9-s + 2.41·11-s − 2.39·25-s − 6.09·43-s − 1.71·49-s − 2.93·67-s + 64/9·81-s + 9.64·99-s + 5.41·107-s + 6.77·113-s − 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 7.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.659151454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.659151454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 2 p T^{2} + 22 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 6 T^{2} + 14 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 46 T^{2} + 862 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 44 T^{2} + 982 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 46 T^{2} + 1126 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 4 T^{2} - 74 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 20 T^{2} + 1558 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 44 T^{2} + 1846 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 4 T^{2} + 3702 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 110 T^{2} + 6342 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 10 T^{2} + 5262 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 148 T^{2} + 12838 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 102 T^{2} + 15254 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 20 T^{2} + 10822 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 364 T^{2} + 51862 T^{4} - 364 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.90464961082538248215340959227, −3.85028593787099717454773798894, −3.68744990110544969960303281942, −3.44070265601804819170473780379, −3.43594872772555167366582573890, −3.40487643327039964204395655511, −3.27461694424472807644676365179, −3.21683267768335035234047326564, −3.06550494943910440800723745661, −2.74863623148288802388242075209, −2.61819394226843485073210836770, −2.41466936823798940392075617990, −2.14434930954531980127187770227, −2.07251972296706838714609929136, −1.94873296647106063895582524594, −1.73366623595429737231555366929, −1.70542610278770498000751589968, −1.57295057595404064724560047944, −1.52773350205451705773781405680, −1.49968759602565959965314114890, −1.09907367160543289958611225446, −1.05606400934635726525198520619, −0.68230312708356509156601453480, −0.52895958142940019983078819058, −0.11285961418822983061052578177,
0.11285961418822983061052578177, 0.52895958142940019983078819058, 0.68230312708356509156601453480, 1.05606400934635726525198520619, 1.09907367160543289958611225446, 1.49968759602565959965314114890, 1.52773350205451705773781405680, 1.57295057595404064724560047944, 1.70542610278770498000751589968, 1.73366623595429737231555366929, 1.94873296647106063895582524594, 2.07251972296706838714609929136, 2.14434930954531980127187770227, 2.41466936823798940392075617990, 2.61819394226843485073210836770, 2.74863623148288802388242075209, 3.06550494943910440800723745661, 3.21683267768335035234047326564, 3.27461694424472807644676365179, 3.40487643327039964204395655511, 3.43594872772555167366582573890, 3.44070265601804819170473780379, 3.68744990110544969960303281942, 3.85028593787099717454773798894, 3.90464961082538248215340959227
Plot not available for L-functions of degree greater than 10.
