| L(s) = 1 | − 3·4-s − 2·5-s + 8·11-s + 6·16-s + 6·19-s + 6·20-s + 25-s + 20·29-s + 28·31-s − 24·44-s + 10·49-s − 16·55-s + 28·59-s + 12·61-s − 10·64-s + 8·71-s − 18·76-s − 44·79-s − 12·80-s + 8·89-s − 12·95-s − 3·100-s + 12·101-s − 8·109-s − 60·116-s + 14·121-s − 84·124-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.894·5-s + 2.41·11-s + 3/2·16-s + 1.37·19-s + 1.34·20-s + 1/5·25-s + 3.71·29-s + 5.02·31-s − 3.61·44-s + 10/7·49-s − 2.15·55-s + 3.64·59-s + 1.53·61-s − 5/4·64-s + 0.949·71-s − 2.06·76-s − 4.95·79-s − 1.34·80-s + 0.847·89-s − 1.23·95-s − 0.299·100-s + 1.19·101-s − 0.766·109-s − 5.57·116-s + 1.27·121-s − 7.54·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.739343190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.739343190\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | \( ( 1 - T )^{6} \) |
| good | 7 | \( 1 - 10 T^{2} + 95 T^{4} - 780 T^{6} + 95 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 42 T^{2} + 815 T^{4} - 13196 T^{6} + 815 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 94 T^{2} + 3999 T^{4} - 108772 T^{6} + 3999 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 10 T + 99 T^{2} - 540 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 14 T + 121 T^{2} - 716 T^{3} + 121 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 50 T^{2} + 1207 T^{4} + 100 T^{6} + 1207 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 107 T^{2} + 16 T^{3} + 107 p T^{4} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 150 T^{2} + 9335 T^{4} - 407060 T^{6} + 9335 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 174 T^{2} + 16271 T^{4} - 934532 T^{6} + 16271 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 146 T^{2} + 10071 T^{4} - 521948 T^{6} + 10071 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 14 T + 229 T^{2} - 1692 T^{3} + 229 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 67 | \( 1 - 274 T^{2} + 36103 T^{4} - 2962972 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 4 T + 133 T^{2} - 504 T^{3} + 133 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 246 T^{2} + 32063 T^{4} - 2771828 T^{6} + 32063 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 22 T + 361 T^{2} + 3676 T^{3} + 361 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 258 T^{2} + 38055 T^{4} - 3905724 T^{6} + 38055 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 4 T + 219 T^{2} - 792 T^{3} + 219 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 186 T^{2} + 22479 T^{4} - 2639468 T^{6} + 22479 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.76288031537317234082263938559, −4.63214684721883093552052388547, −4.55624327140949800230663502313, −4.45638955297933868429435508035, −4.22430529166805014273564020350, −4.00776242079657182479248336068, −3.97337453648983195241681045323, −3.90299463368391312108888821330, −3.82274187862330908256477160804, −3.69689073397363733157244670680, −3.09301493809270337996291917810, −3.07275776770607703734971731954, −2.96064567973377748091679078242, −2.83516342086610340960812765131, −2.71649185425505969210139996932, −2.66680645210095537979187076010, −2.24797646181121992499931123851, −1.85814152080819947033033515446, −1.75178452867041340151318422419, −1.42086723246188977658752398011, −1.18583255619227803978784117068, −0.855264114184596929141082338092, −0.833938879426971351612118464408, −0.70978997633548796566439533630, −0.55394247525803493487218920829,
0.55394247525803493487218920829, 0.70978997633548796566439533630, 0.833938879426971351612118464408, 0.855264114184596929141082338092, 1.18583255619227803978784117068, 1.42086723246188977658752398011, 1.75178452867041340151318422419, 1.85814152080819947033033515446, 2.24797646181121992499931123851, 2.66680645210095537979187076010, 2.71649185425505969210139996932, 2.83516342086610340960812765131, 2.96064567973377748091679078242, 3.07275776770607703734971731954, 3.09301493809270337996291917810, 3.69689073397363733157244670680, 3.82274187862330908256477160804, 3.90299463368391312108888821330, 3.97337453648983195241681045323, 4.00776242079657182479248336068, 4.22430529166805014273564020350, 4.45638955297933868429435508035, 4.55624327140949800230663502313, 4.63214684721883093552052388547, 4.76288031537317234082263938559
Plot not available for L-functions of degree greater than 10.
