| L(s) = 1 | − 4-s + 3·9-s + 2·11-s − 3·16-s − 12·19-s − 2·29-s − 3·36-s − 4·41-s − 2·44-s − 16·59-s − 12·61-s + 3·64-s + 32·71-s + 12·76-s + 18·79-s − 7·81-s + 12·89-s + 6·99-s + 32·101-s − 46·109-s + 2·116-s − 33·121-s + 127-s + 131-s + 137-s + 139-s − 9·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 9-s + 0.603·11-s − 3/4·16-s − 2.75·19-s − 0.371·29-s − 1/2·36-s − 0.624·41-s − 0.301·44-s − 2.08·59-s − 1.53·61-s + 3/8·64-s + 3.79·71-s + 1.37·76-s + 2.02·79-s − 7/9·81-s + 1.27·89-s + 0.603·99-s + 3.18·101-s − 4.40·109-s + 0.185·116-s − 3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3/4·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4645705428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4645705428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 472 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 47 T^{2} + 1024 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3934 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 99 T^{2} + 5912 T^{4} - 99 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 13294 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11734 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 10150 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 9 T + 174 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 135 T^{2} + 14768 T^{4} - 135 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.09566593755943933531611469710, −6.62565912602620124035112007357, −6.41902946282955143987110423168, −6.33564363000852839777055050728, −6.18577323589488734932400121700, −6.03189454618918634418722357521, −5.69265867587008291929648008770, −5.14546886745897441810584416304, −5.02734117768853056709916644152, −4.91832821629773030387593433401, −4.77744983512535875085703793078, −4.40994731801527051968359674453, −4.21199391834556853024353880586, −3.84456070130817693611343974002, −3.77930030276753349828525289941, −3.68620816984602138141416592199, −3.28254681563838282624400359252, −2.67720231338740483210153844531, −2.61575380362978192051842167014, −2.26029920126402067189007420165, −1.97136371300973585458459343513, −1.60962828694916059929954737449, −1.40202066363794967596116420567, −0.833727128339557577617156526727, −0.15957125003530968475800879426,
0.15957125003530968475800879426, 0.833727128339557577617156526727, 1.40202066363794967596116420567, 1.60962828694916059929954737449, 1.97136371300973585458459343513, 2.26029920126402067189007420165, 2.61575380362978192051842167014, 2.67720231338740483210153844531, 3.28254681563838282624400359252, 3.68620816984602138141416592199, 3.77930030276753349828525289941, 3.84456070130817693611343974002, 4.21199391834556853024353880586, 4.40994731801527051968359674453, 4.77744983512535875085703793078, 4.91832821629773030387593433401, 5.02734117768853056709916644152, 5.14546886745897441810584416304, 5.69265867587008291929648008770, 6.03189454618918634418722357521, 6.18577323589488734932400121700, 6.33564363000852839777055050728, 6.41902946282955143987110423168, 6.62565912602620124035112007357, 7.09566593755943933531611469710
