| L(s) = 1 | + 1.35e3·17-s + 676·25-s − 2.31e3·41-s + 4.13e3·49-s + 3.49e4·73-s − 3.64e3·89-s + 2.16e4·97-s + 1.26e4·113-s − 4.23e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.06e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4.67·17-s + 1.08·25-s − 1.37·41-s + 1.72·49-s + 6.55·73-s − 0.459·89-s + 2.30·97-s + 0.991·113-s − 2.89·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.74·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(20.06597848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.06597848\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 338 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 2066 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 21170 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 53474 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 338 T + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 260594 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 23426 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 271726 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 137042 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 3689954 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 578 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 2716850 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4933058 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 9797330 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 22798130 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3794 T + p^{4} T^{2} )^{2}( 1 + 3794 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 28013710 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 32426498 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8734 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 48571438 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 79276558 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 910 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5422 T + p^{4} T^{2} )^{4} \) |
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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
−6.61674474589300111335588745678, −6.05576702489903229388740730686, −5.94686567808535132700306766262, −5.76104086735991138021156678969, −5.55800104919344542821395737093, −5.15311585862818439236687723050, −5.06481134329232952090541124165, −5.00676194609051264333149160386, −4.95725985010806987749937525825, −4.16741354227969547837593114521, −4.06327483745814172307957654558, −3.84302301657577826974050465810, −3.61949135596886600438358171616, −3.33810921391484338915192498561, −3.15158485786315712760027768100, −2.94315850900418685810730755788, −2.71984505465308676734964285526, −2.32645304898405767077708900964, −1.84323732800761486685100729096, −1.73005392832930339684085116787, −1.51265736608282480410452283864, −0.862731231821676076865272957900, −0.801277789202179417450327163821, −0.67827590319561386987674065406, −0.49336405602045809382064856743,
0.49336405602045809382064856743, 0.67827590319561386987674065406, 0.801277789202179417450327163821, 0.862731231821676076865272957900, 1.51265736608282480410452283864, 1.73005392832930339684085116787, 1.84323732800761486685100729096, 2.32645304898405767077708900964, 2.71984505465308676734964285526, 2.94315850900418685810730755788, 3.15158485786315712760027768100, 3.33810921391484338915192498561, 3.61949135596886600438358171616, 3.84302301657577826974050465810, 4.06327483745814172307957654558, 4.16741354227969547837593114521, 4.95725985010806987749937525825, 5.00676194609051264333149160386, 5.06481134329232952090541124165, 5.15311585862818439236687723050, 5.55800104919344542821395737093, 5.76104086735991138021156678969, 5.94686567808535132700306766262, 6.05576702489903229388740730686, 6.61674474589300111335588745678
