| L(s) = 1 | − 4·5-s + 5·9-s + 2·13-s − 8·17-s − 14·25-s + 20·29-s + 8·37-s − 20·45-s + 5·49-s − 4·53-s + 12·61-s − 8·65-s − 84·73-s + 20·81-s + 32·85-s − 18·89-s − 2·97-s − 16·101-s − 24·109-s − 16·113-s + 10·117-s + 13·121-s + 80·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 5/3·9-s + 0.554·13-s − 1.94·17-s − 2.79·25-s + 3.71·29-s + 1.31·37-s − 2.98·45-s + 5/7·49-s − 0.549·53-s + 1.53·61-s − 0.992·65-s − 9.83·73-s + 20/9·81-s + 3.47·85-s − 1.90·89-s − 0.203·97-s − 1.59·101-s − 2.29·109-s − 1.50·113-s + 0.924·117-s + 1.18·121-s + 7.15·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9539210865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9539210865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( 1 - 5 T^{2} + 5 T^{4} - 10 T^{6} + 94 T^{8} - 10 p^{2} T^{10} + 5 p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 7 | \( 1 - 5 T^{2} - 75 T^{4} - 10 T^{6} + 6374 T^{8} - 10 p^{2} T^{10} - 75 p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 13 T^{2} + 93 T^{4} + 2158 T^{6} - 29314 T^{8} + 2158 p^{2} T^{10} + 93 p^{4} T^{12} - 13 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 4 T - 5 T^{2} - 52 T^{3} - 120 T^{4} - 52 p T^{5} - 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 3 T^{2} - 507 T^{4} - 618 T^{6} + 132686 T^{8} - 618 p^{2} T^{10} - 507 p^{4} T^{12} + 3 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 29 T^{2} - 83 T^{4} + 3886 T^{6} + 129046 T^{8} + 3886 p^{2} T^{10} - 83 p^{4} T^{12} - 29 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 68 T^{2} + 2806 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T - 45 T^{2} + 52 T^{3} + 1760 T^{4} + 52 p T^{5} - 45 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 21 T + 188 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2}( 1 + 21 T + 188 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( 1 - 101 T^{2} + 5181 T^{4} - 133522 T^{6} + 2919950 T^{8} - 133522 p^{2} T^{10} + 5181 p^{4} T^{12} - 101 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 72 T^{2} + 2382 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + T + 102 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 - 21 T^{2} - 6627 T^{4} - 2226 T^{6} + 36304142 T^{8} - 2226 p^{2} T^{10} - 6627 p^{4} T^{12} - 21 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 6 T - 27 T^{2} + 354 T^{3} - 1948 T^{4} + 354 p T^{5} - 27 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 149 T^{2} + 8901 T^{4} - 643978 T^{6} + 57078590 T^{8} - 643978 p^{2} T^{10} + 8901 p^{4} T^{12} - 149 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 205 T^{2} + 21645 T^{4} - 2111090 T^{6} + 178084694 T^{8} - 2111090 p^{2} T^{10} + 21645 p^{4} T^{12} - 205 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 21 T + 218 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 64 T^{2} + 10174 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 252 T^{2} + 28566 T^{4} + 252 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 9 T - 79 T^{2} - 162 T^{3} + 10470 T^{4} - 162 p T^{5} - 79 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + T - 87 T^{2} - 106 T^{3} - 1762 T^{4} - 106 p T^{5} - 87 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.70628281085547355696063768583, −4.69912647550065897630110722292, −4.63227318732739165860121632332, −4.55165130598770223806425384934, −4.39706115426976688268827237943, −4.21216447585704202026555726346, −4.13999949445329229275002042388, −4.10461914382757405090503890863, −3.84897654514970693305286304510, −3.65538940769026390032448938708, −3.50167979350840205087234374235, −3.41423376194738707303960877100, −3.27440240414119681047159960488, −2.80161728634387450010690245087, −2.76859461347779009020909261489, −2.60188210447576842164962080250, −2.51573433466867128932413780597, −2.40986973029628232472198183716, −1.88801661508749921013684767836, −1.78643903968965349693797247115, −1.42707239442534010509793832199, −1.37853139296822822563482489152, −1.26902293301586231921026772178, −0.54194731449038268215551442136, −0.27654549078756125280518451938,
0.27654549078756125280518451938, 0.54194731449038268215551442136, 1.26902293301586231921026772178, 1.37853139296822822563482489152, 1.42707239442534010509793832199, 1.78643903968965349693797247115, 1.88801661508749921013684767836, 2.40986973029628232472198183716, 2.51573433466867128932413780597, 2.60188210447576842164962080250, 2.76859461347779009020909261489, 2.80161728634387450010690245087, 3.27440240414119681047159960488, 3.41423376194738707303960877100, 3.50167979350840205087234374235, 3.65538940769026390032448938708, 3.84897654514970693305286304510, 4.10461914382757405090503890863, 4.13999949445329229275002042388, 4.21216447585704202026555726346, 4.39706115426976688268827237943, 4.55165130598770223806425384934, 4.63227318732739165860121632332, 4.69912647550065897630110722292, 4.70628281085547355696063768583
Plot not available for L-functions of degree greater than 10.
