| L(s) = 1 | + 8·5-s − 8·9-s + 6·13-s − 4·17-s + 24·25-s + 12·29-s + 24·37-s + 20·41-s − 64·45-s − 22·49-s + 32·53-s + 8·61-s + 48·65-s − 12·73-s + 24·81-s − 32·85-s − 4·89-s + 20·97-s + 48·101-s + 24·109-s − 24·113-s − 48·117-s − 24·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 8/3·9-s + 1.66·13-s − 0.970·17-s + 24/5·25-s + 2.22·29-s + 3.94·37-s + 3.12·41-s − 9.54·45-s − 3.14·49-s + 4.39·53-s + 1.02·61-s + 5.95·65-s − 1.40·73-s + 8/3·81-s − 3.47·85-s − 0.423·89-s + 2.03·97-s + 4.77·101-s + 2.29·109-s − 2.25·113-s − 4.43·117-s − 2.18·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(29.88998086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.88998086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 - T )^{6} \) |
| good | 3 | \( 1 + 8 T^{2} + 40 T^{4} + 142 T^{6} + 40 p^{2} T^{8} + 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( ( 1 - 4 T + 12 T^{2} - 6 p T^{3} + 12 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 + 22 T^{2} + 300 T^{4} + 2488 T^{6} + 300 p^{2} T^{8} + 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 + 24 T^{2} + 383 T^{4} + 4480 T^{6} + 383 p^{2} T^{8} + 24 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 2 T + 28 T^{2} + 24 T^{3} + 28 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 + 48 T^{2} + 1103 T^{4} + 20464 T^{6} + 1103 p^{2} T^{8} + 48 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 + 98 T^{2} + 4655 T^{4} + 134268 T^{6} + 4655 p^{2} T^{8} + 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 31 | \( 1 + 152 T^{2} + 10487 T^{4} + 417504 T^{6} + 10487 p^{2} T^{8} + 152 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 12 T + 116 T^{2} - 838 T^{3} + 116 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 - 10 T + 59 T^{2} - 308 T^{3} + 59 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 224 T^{2} + 22152 T^{4} + 1235702 T^{6} + 22152 p^{2} T^{8} + 224 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 + 230 T^{2} + 24028 T^{4} + 1450088 T^{6} + 24028 p^{2} T^{8} + 230 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 16 T + 147 T^{2} - 1104 T^{3} + 147 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 224 T^{2} + 22687 T^{4} + 1526288 T^{6} + 22687 p^{2} T^{8} + 224 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 4 T + 87 T^{2} - 616 T^{3} + 87 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 + 200 T^{2} + 19887 T^{4} + 1426976 T^{6} + 19887 p^{2} T^{8} + 200 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 + 230 T^{2} + 28604 T^{4} + 2451288 T^{6} + 28604 p^{2} T^{8} + 230 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 6 T + 59 T^{2} + 1004 T^{3} + 59 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 202 T^{2} + 30399 T^{4} + 2693164 T^{6} + 30399 p^{2} T^{8} + 202 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( 1 + 280 T^{2} + 44047 T^{4} + 4436224 T^{6} + 44047 p^{2} T^{8} + 280 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 2 T + 235 T^{2} + 324 T^{3} + 235 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - 10 T + 191 T^{2} - 1452 T^{3} + 191 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.53686904309086578594722414429, −4.36412772063534031305024219901, −4.20565265490568762704411645251, −3.95940743365128932279339145843, −3.82022617313923552110811879490, −3.80258807302052414823093745020, −3.68711389907220452100021290454, −3.13393754920328011202412473763, −3.07410319842989738819667477550, −3.05190695585005502433611463441, −2.93362146801560989111058666471, −2.71897092013540772581794935296, −2.65809232429764797812024094935, −2.45478937191893209061591099051, −2.35131744514975240519397387510, −2.00120674114947861340266820837, −1.96470195727427781537973493651, −1.89673752711720493933666387124, −1.89161096189450299477721075369, −1.29386475901641546649857948916, −1.25984538120736786829604278181, −0.860069885610132711026043451988, −0.800410563065936705852700341572, −0.57397808760970688918900023380, −0.46933103496226217639687674695,
0.46933103496226217639687674695, 0.57397808760970688918900023380, 0.800410563065936705852700341572, 0.860069885610132711026043451988, 1.25984538120736786829604278181, 1.29386475901641546649857948916, 1.89161096189450299477721075369, 1.89673752711720493933666387124, 1.96470195727427781537973493651, 2.00120674114947861340266820837, 2.35131744514975240519397387510, 2.45478937191893209061591099051, 2.65809232429764797812024094935, 2.71897092013540772581794935296, 2.93362146801560989111058666471, 3.05190695585005502433611463441, 3.07410319842989738819667477550, 3.13393754920328011202412473763, 3.68711389907220452100021290454, 3.80258807302052414823093745020, 3.82022617313923552110811879490, 3.95940743365128932279339145843, 4.20565265490568762704411645251, 4.36412772063534031305024219901, 4.53686904309086578594722414429
Plot not available for L-functions of degree greater than 10.
