| L(s) = 1 | + 6·2-s + 21·4-s + 4·5-s + 4·7-s + 56·8-s + 4·9-s + 24·10-s + 24·14-s + 126·16-s + 24·18-s + 84·20-s + 16·25-s + 84·28-s + 4·29-s + 252·32-s + 16·35-s + 84·36-s + 16·37-s + 224·40-s + 16·45-s + 20·47-s + 96·50-s + 224·56-s + 24·58-s + 20·61-s + 16·63-s + 462·64-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 21/2·4-s + 1.78·5-s + 1.51·7-s + 19.7·8-s + 4/3·9-s + 7.58·10-s + 6.41·14-s + 63/2·16-s + 5.65·18-s + 18.7·20-s + 16/5·25-s + 15.8·28-s + 0.742·29-s + 44.5·32-s + 2.70·35-s + 14·36-s + 2.63·37-s + 35.4·40-s + 2.38·45-s + 2.91·47-s + 13.5·50-s + 29.9·56-s + 3.15·58-s + 2.56·61-s + 2.01·63-s + 57.7·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(551.1203597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(551.1203597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{6} \) |
| 5 | \( 1 - 4 T + 18 T^{3} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 4 T^{2} + 16 T^{4} - 62 T^{6} + 16 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 2 T + 6 T^{2} - 8 T^{3} + 6 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{3} \) |
| 17 | \( 1 + 752 T^{4} - 50 T^{6} + 752 p^{2} T^{8} + p^{6} T^{12} \) |
| 19 | \( 1 - 22 T^{2} + 199 T^{4} - 3796 T^{6} + 199 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 70 T^{2} + 2127 T^{4} - 47860 T^{6} + 2127 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 43 T^{2} - 156 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 82 T^{2} + 3839 T^{4} - 133596 T^{6} + 3839 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 8 T + 112 T^{2} - 590 T^{3} + 112 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 2 p T^{2} + 5999 T^{4} - 251676 T^{6} + 5999 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 84 T^{2} + 5136 T^{4} - 202462 T^{6} + 5136 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 10 T + 158 T^{2} - 932 T^{3} + 158 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 250 T^{2} + 28167 T^{4} - 1878700 T^{6} + 28167 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 262 T^{2} + 32279 T^{4} - 2394036 T^{6} + 32279 p^{2} T^{8} - 262 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 10 T + 135 T^{2} - 1252 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 + 12 T + 221 T^{2} + 1528 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 - 292 T^{2} + 41984 T^{4} - 3693606 T^{6} + 41984 p^{2} T^{8} - 292 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 6 T + p T^{2} )^{6} \) |
| 79 | \( ( 1 + 28 T + 453 T^{2} + 4744 T^{3} + 453 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 16 T + 289 T^{2} - 2496 T^{3} + 289 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 442 T^{2} + 87839 T^{4} - 10041516 T^{6} + 87839 p^{2} T^{8} - 442 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 26 T + 343 T^{2} + 3452 T^{3} + 343 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.85855478493537862800837925862, −4.71129253646843908611369572251, −4.51688806885238980277861219171, −4.49334550823596115449090682909, −4.24783197430843768694668735353, −4.20600733144575485069778031388, −4.15273436518531616801236207524, −3.97432096781888604781625691582, −3.83504363464539664999088389411, −3.56696209125989280665753961778, −3.28479080439969732131423518110, −3.06353779930961540572044933726, −2.82836891049009189456424363647, −2.79623216542306871698600483267, −2.72569933348782284009343818379, −2.72367220498407300921208038530, −2.34653198462494855883036181860, −2.15474486956818650190726832953, −1.99276106658397226537335992230, −1.57078352798726071433316745592, −1.52815895722685042948688064492, −1.48746621614121814174246702391, −1.30019465767275469182866636123, −0.960156514934724811617455245841, −0.66410200954421012276281290683,
0.66410200954421012276281290683, 0.960156514934724811617455245841, 1.30019465767275469182866636123, 1.48746621614121814174246702391, 1.52815895722685042948688064492, 1.57078352798726071433316745592, 1.99276106658397226537335992230, 2.15474486956818650190726832953, 2.34653198462494855883036181860, 2.72367220498407300921208038530, 2.72569933348782284009343818379, 2.79623216542306871698600483267, 2.82836891049009189456424363647, 3.06353779930961540572044933726, 3.28479080439969732131423518110, 3.56696209125989280665753961778, 3.83504363464539664999088389411, 3.97432096781888604781625691582, 4.15273436518531616801236207524, 4.20600733144575485069778031388, 4.24783197430843768694668735353, 4.49334550823596115449090682909, 4.51688806885238980277861219171, 4.71129253646843908611369572251, 4.85855478493537862800837925862
Plot not available for L-functions of degree greater than 10.
