| L(s) = 1 | + 8·19-s − 16·25-s + 32·43-s + 32·49-s + 32·67-s + 16·73-s − 8·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 1.83·19-s − 3.19·25-s + 4.87·43-s + 32/7·49-s + 3.90·67-s + 1.87·73-s − 0.812·97-s + 4·121-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 + 2.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.529951505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.529951505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 8 T^{2} + 33 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 16 T^{2} + 129 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 32 T^{2} + 1182 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 80 T^{2} + 3150 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 64 T^{2} + 2913 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 32 T^{2} + 1806 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 88 T^{2} + 4110 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 44 T^{2} + 2790 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 176 T^{2} + 13065 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 100 T^{2} + 7830 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 + 176 T^{2} + 16638 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4 T + 117 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 256 T^{2} + 28734 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 214 T^{2} + 23115 T^{4} - 214 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 52 T^{2} - 2490 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + T + p T^{2} )^{8} \) |
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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
−4.26336714120315615373162669209, −4.22591597986770543772465928908, −4.03770289616017545775363228232, −3.93412429774273470551505652582, −3.86702694515791834322844766213, −3.71252888916599222679239128433, −3.65117814155207515853288012712, −3.62125385398019138086298784702, −3.45022242313395660766719518460, −2.98818634752904705501888255725, −2.96049425193119500130518239167, −2.76891770679206037721418099139, −2.70652671464320318312253836531, −2.53210870945889866503666680983, −2.39032931511916959826561136391, −2.20055732850667111577178426512, −2.16298109142292269839949396496, −1.89294673172173153945813497113, −1.72786835695177870479669570346, −1.58058285366032570293682498013, −1.13770639634418469473904801268, −1.02597886187888574995064217013, −0.74313374750953099124489877589, −0.57522832837880327525537985606, −0.51772943518767081597301335635,
0.51772943518767081597301335635, 0.57522832837880327525537985606, 0.74313374750953099124489877589, 1.02597886187888574995064217013, 1.13770639634418469473904801268, 1.58058285366032570293682498013, 1.72786835695177870479669570346, 1.89294673172173153945813497113, 2.16298109142292269839949396496, 2.20055732850667111577178426512, 2.39032931511916959826561136391, 2.53210870945889866503666680983, 2.70652671464320318312253836531, 2.76891770679206037721418099139, 2.96049425193119500130518239167, 2.98818634752904705501888255725, 3.45022242313395660766719518460, 3.62125385398019138086298784702, 3.65117814155207515853288012712, 3.71252888916599222679239128433, 3.86702694515791834322844766213, 3.93412429774273470551505652582, 4.03770289616017545775363228232, 4.22591597986770543772465928908, 4.26336714120315615373162669209
Plot not available for L-functions of degree greater than 10.
