| L(s) = 1 | − 8·7-s + 4·9-s + 4·25-s + 32·31-s + 34·49-s − 32·63-s − 6·81-s + 16·103-s − 72·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 4/3·9-s + 4/5·25-s + 5.74·31-s + 34/7·49-s − 4.03·63-s − 2/3·81-s + 1.57·103-s − 6.77·113-s − 1.81·121-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 − 2.15·169-s + 0.0760·173-s − 2.41·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.652704028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.652704028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.54769846984708265365893326726, −6.38964646932327650238454278167, −6.30156344119126328397286120301, −6.25374729537269986923632495007, −5.96126094892780603151314100302, −5.48951149048810796294566268324, −5.28706318632413139809969280326, −5.16926717120015883289475225174, −4.86283426764271311878706954171, −4.53069516972541974115031320502, −4.33767963487324340621380104902, −4.24323455797769733300531422369, −3.96363554603044448416685189462, −3.62145335985329293199719419578, −3.61045470272336024425967764226, −3.03998442316696008251800990983, −2.89476893826406124826326494463, −2.72134195976213694686958656727, −2.64670560181074111228044535550, −2.44313757805799272978955123939, −1.68364349675022745156352715638, −1.44704178786612253807773499313, −1.07709711449212725480744468424, −0.69235760161598458308861321334, −0.40463398451750074972246008940,
0.40463398451750074972246008940, 0.69235760161598458308861321334, 1.07709711449212725480744468424, 1.44704178786612253807773499313, 1.68364349675022745156352715638, 2.44313757805799272978955123939, 2.64670560181074111228044535550, 2.72134195976213694686958656727, 2.89476893826406124826326494463, 3.03998442316696008251800990983, 3.61045470272336024425967764226, 3.62145335985329293199719419578, 3.96363554603044448416685189462, 4.24323455797769733300531422369, 4.33767963487324340621380104902, 4.53069516972541974115031320502, 4.86283426764271311878706954171, 5.16926717120015883289475225174, 5.28706318632413139809969280326, 5.48951149048810796294566268324, 5.96126094892780603151314100302, 6.25374729537269986923632495007, 6.30156344119126328397286120301, 6.38964646932327650238454278167, 6.54769846984708265365893326726
