L(s) = 1 | − 2·4-s − 6·9-s − 8·11-s + 16-s + 8·29-s + 24·31-s + 12·36-s − 40·41-s + 16·44-s − 10·49-s − 16·59-s + 12·61-s + 18·64-s − 32·71-s + 48·79-s + 19·81-s − 12·89-s + 48·99-s − 4·101-s − 20·109-s − 16·116-s + 52·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 4-s − 2·9-s − 2.41·11-s + 1/4·16-s + 1.48·29-s + 4.31·31-s + 2·36-s − 6.24·41-s + 2.41·44-s − 1.42·49-s − 2.08·59-s + 1.53·61-s + 9/4·64-s − 3.79·71-s + 5.40·79-s + 19/9·81-s − 1.27·89-s + 4.82·99-s − 0.398·101-s − 1.91·109-s − 1.48·116-s + 4.72·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3825312043\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3825312043\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
good | 2 | \( 1 + p T^{2} + 3 T^{4} - 7 p T^{6} - 31 T^{8} - 7 p^{3} T^{10} + 3 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} \) |
| 3 | \( 1 + 2 p T^{2} + 17 T^{4} + 2 p T^{6} - 44 T^{8} + 2 p^{3} T^{10} + 17 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 16 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 44 T^{2} + 1002 T^{4} + 15664 T^{6} + 229331 T^{8} + 15664 p^{2} T^{10} + 1002 p^{4} T^{12} + 44 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 86 T^{2} + 4497 T^{4} + 158326 T^{6} + 4211876 T^{8} + 158326 p^{2} T^{10} + 4497 p^{4} T^{12} + 86 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - T + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 118 T^{2} + 6979 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 90 T^{2} + 5891 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 164 T^{2} + 15066 T^{4} + 1018768 T^{6} + 56507555 T^{8} + 1018768 p^{2} T^{10} + 15066 p^{4} T^{12} + 164 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 8 T + 2 T^{2} - 448 T^{3} - 3413 T^{4} - 448 p T^{5} + 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 6 T - 23 T^{2} + 378 T^{3} - 2436 T^{4} + 378 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 22 T^{2} - 7647 T^{4} - 18634 T^{6} + 43789364 T^{8} - 18634 p^{2} T^{10} - 7647 p^{4} T^{12} + 22 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 268 T^{2} + 43338 T^{4} + 4777904 T^{6} + 400303859 T^{8} + 4777904 p^{2} T^{10} + 43338 p^{4} T^{12} + 268 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 24 T + 282 T^{2} - 3264 T^{3} + 34691 T^{4} - 3264 p T^{5} + 282 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 6 T^{2} + 13139 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 6 T - 119 T^{2} - 138 T^{3} + 12900 T^{4} - 138 p T^{5} - 119 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 252 T^{2} + 30086 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−5.82896304826404804420291010218, −5.62337573717000565981452316681, −5.55820415894624284316431430291, −5.25666793200693757427126410340, −5.13931629709599918597248498819, −4.95037178070989687989230288398, −4.92547783178839153430345347150, −4.80664157525126100071759590384, −4.72332895102875896868947592541, −4.42187856215660411374569361462, −4.37273869551984358665684694296, −4.02503856356081841940572454110, −3.66778659466368971258730215142, −3.52394600398808080791863008330, −3.45056157374777453049629116384, −3.02878436875264574413521983813, −2.99199631447026265722247462094, −2.97673960170604164116489493613, −2.61257128336246922012246576871, −2.55406100416615270122480528761, −2.13208403683115792993625752563, −1.83529524926067865821942978040, −1.59954910400943847851394369035, −0.899615210585052646956493991256, −0.34496457821810422582522828441,
0.34496457821810422582522828441, 0.899615210585052646956493991256, 1.59954910400943847851394369035, 1.83529524926067865821942978040, 2.13208403683115792993625752563, 2.55406100416615270122480528761, 2.61257128336246922012246576871, 2.97673960170604164116489493613, 2.99199631447026265722247462094, 3.02878436875264574413521983813, 3.45056157374777453049629116384, 3.52394600398808080791863008330, 3.66778659466368971258730215142, 4.02503856356081841940572454110, 4.37273869551984358665684694296, 4.42187856215660411374569361462, 4.72332895102875896868947592541, 4.80664157525126100071759590384, 4.92547783178839153430345347150, 4.95037178070989687989230288398, 5.13931629709599918597248498819, 5.25666793200693757427126410340, 5.55820415894624284316431430291, 5.62337573717000565981452316681, 5.82896304826404804420291010218
Plot not available for L-functions of degree greater than 10.