| L(s) = 1 | + 4·3-s + 2·4-s + 10·9-s + 8·12-s + 3·16-s − 8·17-s + 16·23-s + 4·25-s + 20·27-s + 8·29-s + 20·36-s − 16·43-s + 12·48-s + 12·49-s − 32·51-s − 8·53-s + 8·61-s + 12·64-s − 16·68-s + 64·69-s + 16·75-s + 35·81-s + 32·87-s + 32·92-s + 8·100-s − 8·101-s − 32·103-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 4-s + 10/3·9-s + 2.30·12-s + 3/4·16-s − 1.94·17-s + 3.33·23-s + 4/5·25-s + 3.84·27-s + 1.48·29-s + 10/3·36-s − 2.43·43-s + 1.73·48-s + 12/7·49-s − 4.48·51-s − 1.09·53-s + 1.02·61-s + 3/2·64-s − 1.94·68-s + 7.70·69-s + 1.84·75-s + 35/9·81-s + 3.43·87-s + 3.33·92-s + 4/5·100-s − 0.796·101-s − 3.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.932304504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.932304504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 1414 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13174 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 20566 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 12134 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 30166 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 11910 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.130782279648337196236388229706, −7.39849352577358099294745336808, −7.38483113781490840572158275796, −7.16113606393434282866593448667, −7.15576024923931373735185368729, −6.62462002213000556669800919970, −6.60311533599608811756224199124, −6.52255265429510025199285516384, −6.10226728422057323485812251585, −5.60603661368182320709479726436, −5.30581934057749189589258873043, −4.94758832672999984264911150930, −4.71066652253955467134733220451, −4.70352466768922871050771490419, −4.26097734204410416675861396163, −3.79239884410095802067850699850, −3.54308177137946858808384954457, −3.38869672699341469396418259357, −2.79596718281181858588101229083, −2.75531128747965607338796488580, −2.65700944160400447192490663477, −2.15222466683602588959249188618, −1.80791716948160447606819727956, −1.29298295527238399051943004688, −0.935288527558163476420023430515,
0.935288527558163476420023430515, 1.29298295527238399051943004688, 1.80791716948160447606819727956, 2.15222466683602588959249188618, 2.65700944160400447192490663477, 2.75531128747965607338796488580, 2.79596718281181858588101229083, 3.38869672699341469396418259357, 3.54308177137946858808384954457, 3.79239884410095802067850699850, 4.26097734204410416675861396163, 4.70352466768922871050771490419, 4.71066652253955467134733220451, 4.94758832672999984264911150930, 5.30581934057749189589258873043, 5.60603661368182320709479726436, 6.10226728422057323485812251585, 6.52255265429510025199285516384, 6.60311533599608811756224199124, 6.62462002213000556669800919970, 7.15576024923931373735185368729, 7.16113606393434282866593448667, 7.38483113781490840572158275796, 7.39849352577358099294745336808, 8.130782279648337196236388229706
