| L(s) = 1 | + 2·3-s + 2·4-s + 2·5-s − 2·7-s + 9-s − 4·11-s + 4·12-s − 12·13-s + 4·15-s + 4·16-s + 4·17-s + 2·19-s + 4·20-s − 4·21-s + 4·23-s + 25-s − 2·27-s − 4·28-s + 16·29-s + 6·31-s − 8·33-s − 4·35-s + 2·36-s + 14·37-s − 24·39-s − 8·41-s − 36·43-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 4-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.15·12-s − 3.32·13-s + 1.03·15-s + 16-s + 0.970·17-s + 0.458·19-s + 0.894·20-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.384·27-s − 0.755·28-s + 2.97·29-s + 1.07·31-s − 1.39·33-s − 0.676·35-s + 1/3·36-s + 2.30·37-s − 3.84·39-s − 1.24·41-s − 5.48·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.653919374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653919374\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 6 T + 33 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 56 p T^{5} - 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 62 T^{3} - 388 T^{4} + 62 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T - 16 T^{2} + 56 T^{3} + 127 T^{4} + 56 p T^{5} - 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 6 T - 27 T^{2} - 6 T^{3} + 1892 T^{4} - 6 p T^{5} - 27 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 14 T + 75 T^{2} - 658 T^{3} + 6020 T^{4} - 658 p T^{5} + 75 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 18 T + 165 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T + 46 T^{2} + 496 T^{3} - 2061 T^{4} + 496 p T^{5} + 46 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 8 T - 50 T^{2} + 64 T^{3} + 6795 T^{4} + 64 p T^{5} - 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 448 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T + 31 T^{2} + 322 T^{3} - 4028 T^{4} + 322 p T^{5} + 31 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 10 T - 21 T^{2} + 250 T^{3} + 2012 T^{4} + 250 p T^{5} - 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 2 T - 123 T^{2} - 62 T^{3} + 9572 T^{4} - 62 p T^{5} - 123 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 180 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16 T + 32 T^{2} - 736 T^{3} + 19471 T^{4} - 736 p T^{5} + 32 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 16 T + 250 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−10.12979718346965109736494314099, −9.810283539703820155971404554952, −9.629025320752974606142441286694, −9.582192795867821887993136905510, −9.319754419749770283371809151912, −8.562070783046229850934412225092, −8.370373083576000785921806442735, −8.023002158108103590266103382858, −7.916902001221513787425387266516, −7.74998069104029779940496067301, −7.03228571844964354048694761145, −6.91256119582870925137041749674, −6.80204417439784926319119941577, −6.26845592511871665743518898927, −6.11473449179698966373873177186, −5.29006842559240291694108352725, −5.10864985383856050819856580269, −5.02312112477664554148778556602, −4.67396113575950758188924115996, −3.82006576500631858148861929335, −3.14266139598966056537513697754, −2.91151709495536626609309891081, −2.60443631793151104599636944450, −2.52482421687637334071804695949, −1.57664261147576100464138211758,
1.57664261147576100464138211758, 2.52482421687637334071804695949, 2.60443631793151104599636944450, 2.91151709495536626609309891081, 3.14266139598966056537513697754, 3.82006576500631858148861929335, 4.67396113575950758188924115996, 5.02312112477664554148778556602, 5.10864985383856050819856580269, 5.29006842559240291694108352725, 6.11473449179698966373873177186, 6.26845592511871665743518898927, 6.80204417439784926319119941577, 6.91256119582870925137041749674, 7.03228571844964354048694761145, 7.74998069104029779940496067301, 7.916902001221513787425387266516, 8.023002158108103590266103382858, 8.370373083576000785921806442735, 8.562070783046229850934412225092, 9.319754419749770283371809151912, 9.582192795867821887993136905510, 9.629025320752974606142441286694, 9.810283539703820155971404554952, 10.12979718346965109736494314099
