L(s) = 1 | + 2·9-s + 12·17-s − 10·25-s + 12·41-s − 14·49-s + 4·73-s − 5·81-s + 36·89-s − 20·97-s + 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 2/3·9-s + 2.91·17-s − 2·25-s + 1.87·41-s − 2·49-s + 0.468·73-s − 5/9·81-s + 3.81·89-s − 2.03·97-s + 3.38·113-s − 1.27·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.586248366\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.586248366\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−12.35797581564216331802133964919, −11.75655140830289064450817435587, −11.43416050332192748416656544847, −10.80930439699301989240452536246, −10.13890570115710586872097299926, −9.938036229146395142845664532292, −9.536453957000876163766901881878, −9.033934658635621728990784548147, −8.066335838706079171976710141123, −7.85204783264984617615072264980, −7.55299395754220445658708645745, −6.84371134663949782422454400794, −6.00140065324702785753059105988, −5.80484397546847799453205431959, −5.09546563559245222823656645910, −4.39774330912317846963186570778, −3.64183884460257760595955162875, −3.22834670546030511963807300880, −2.09259386172801032688627778191, −1.13265678082836875231989401788,
1.13265678082836875231989401788, 2.09259386172801032688627778191, 3.22834670546030511963807300880, 3.64183884460257760595955162875, 4.39774330912317846963186570778, 5.09546563559245222823656645910, 5.80484397546847799453205431959, 6.00140065324702785753059105988, 6.84371134663949782422454400794, 7.55299395754220445658708645745, 7.85204783264984617615072264980, 8.066335838706079171976710141123, 9.033934658635621728990784548147, 9.536453957000876163766901881878, 9.938036229146395142845664532292, 10.13890570115710586872097299926, 10.80930439699301989240452536246, 11.43416050332192748416656544847, 11.75655140830289064450817435587, 12.35797581564216331802133964919