| L(s) = 1 | + 2·2-s − 4-s + 2·5-s − 4·8-s + 9-s + 4·10-s − 4·11-s + 6·13-s + 2·18-s − 4·19-s − 2·20-s − 8·22-s + 18·23-s + 2·25-s + 12·26-s + 14·29-s − 20·31-s + 2·32-s − 36-s + 10·37-s − 8·38-s − 8·40-s + 20·41-s + 4·44-s + 2·45-s + 36·46-s − 24·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 0.894·5-s − 1.41·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s + 1.66·13-s + 0.471·18-s − 0.917·19-s − 0.447·20-s − 1.70·22-s + 3.75·23-s + 2/5·25-s + 2.35·26-s + 2.59·29-s − 3.59·31-s + 0.353·32-s − 1/6·36-s + 1.64·37-s − 1.29·38-s − 1.26·40-s + 3.12·41-s + 0.603·44-s + 0.298·45-s + 5.30·46-s − 3.50·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.771414017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.771414017\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + 5 T^{2} - p^{3} T^{3} + 13 T^{4} - p^{4} T^{5} + 5 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 8 T^{3} + 31 T^{4} - 8 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 20 T^{2} + 76 T^{3} + 247 T^{4} + 76 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} - 120 T^{3} - 601 T^{4} - 120 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 18 T + 180 T^{2} - 1296 T^{3} + 7139 T^{4} - 1296 p T^{5} + 180 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1500 T^{3} + 9314 T^{4} + 1500 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 10 T + 125 T^{2} - 870 T^{3} + 6656 T^{4} - 870 p T^{5} + 125 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 20 T + 221 T^{2} - 1688 T^{3} + 11356 T^{4} - 1688 p T^{5} + 221 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \) |
| 47 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2280 T^{3} + 15746 T^{4} + 2280 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 139 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10 T + 26 T^{2} + 476 T^{3} - 6545 T^{4} + 476 p T^{5} + 26 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 173 T^{2} - 1500 T^{3} + 14832 T^{4} - 1500 p T^{5} + 173 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 18 T + 90 T^{2} - 1212 T^{3} - 18625 T^{4} - 1212 p T^{5} + 90 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 8 T + 32 T^{2} - 880 T^{3} - 8561 T^{4} - 880 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} + 96 T^{3} - 10033 T^{4} + 96 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 8878 T^{4} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 22 T + 170 T^{2} - 68 T^{3} - 10433 T^{4} - 68 p T^{5} + 170 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 10 T + 26 T^{2} - 780 T^{3} - 14449 T^{4} - 780 p T^{5} + 26 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.710119611971559169287174154162, −8.595412920257115749384439347850, −8.160817116057224201840662263372, −7.956193887299941229098777127405, −7.49702213323867929922724178135, −7.48301320160365157939685343255, −6.88652062083032712304779561488, −6.76894407315084252016827220238, −6.55913491159693394578312649433, −6.08431315161418920363579079618, −5.98405662877678639751543767251, −5.44585522848869558145179753428, −5.36988408368673516181525594779, −5.05611033501172210893795512393, −4.94546248510286264633523335677, −4.61200138536940701450318660164, −4.18888201694351820522358549025, −3.99595142084720817810288466031, −3.73373233816265746266074072642, −3.19931871238343945130541265753, −2.90226811520123641328650619758, −2.62187921511745008368408118346, −2.07633351458898133562482952634, −1.31347903059825464755341399188, −0.910902436428917534686185350642,
0.910902436428917534686185350642, 1.31347903059825464755341399188, 2.07633351458898133562482952634, 2.62187921511745008368408118346, 2.90226811520123641328650619758, 3.19931871238343945130541265753, 3.73373233816265746266074072642, 3.99595142084720817810288466031, 4.18888201694351820522358549025, 4.61200138536940701450318660164, 4.94546248510286264633523335677, 5.05611033501172210893795512393, 5.36988408368673516181525594779, 5.44585522848869558145179753428, 5.98405662877678639751543767251, 6.08431315161418920363579079618, 6.55913491159693394578312649433, 6.76894407315084252016827220238, 6.88652062083032712304779561488, 7.48301320160365157939685343255, 7.49702213323867929922724178135, 7.956193887299941229098777127405, 8.160817116057224201840662263372, 8.595412920257115749384439347850, 8.710119611971559169287174154162
