| L(s) = 1 | + 8·11-s − 4·13-s − 5·16-s − 8·19-s − 20·23-s − 14·25-s + 8·41-s − 8·43-s − 8·49-s + 24·67-s + 36·71-s − 48·103-s + 12·107-s + 32·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 1.10·13-s − 5/4·16-s − 1.83·19-s − 4.17·23-s − 2.79·25-s + 1.24·41-s − 1.21·43-s − 8/7·49-s + 2.93·67-s + 4.27·71-s − 4.72·103-s + 1.16·107-s + 3.01·113-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.38·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.092433350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.092433350\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 5 T^{4} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 8 T^{2} + 66 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 4 T^{2} + 1158 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 40 T^{2} + 3090 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 87 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4370 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 80 T^{2} + 3330 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 176 T^{2} + 14274 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 200 T^{2} + 17010 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 167 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 18 T + 211 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 232 T^{2} + 23682 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 152 T^{2} + 17826 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 180 T^{2} + 17030 T^{4} + 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 56 T^{2} + 19554 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.54891211130720804574207155006, −5.41248176701585342978368169148, −5.16390021798114330520574733130, −4.87951498259642064143071842277, −4.80633183370004306612720727497, −4.30072384886374500052138437362, −4.27624149011911824441656667614, −4.19787940702584306602654924709, −4.11175821591785771036656512434, −3.78722775984697590495323101394, −3.74623448102970239794614645092, −3.73787013392988393287581521115, −3.26601454483731431835599146032, −2.94473902732425550788567762622, −2.85249728577045923299528853404, −2.27248337529066737729467379761, −2.19988385926848787495153851337, −2.06654955129862165083267573078, −2.03141536428772317914521004931, −1.84113175081442612585096897771, −1.56617192724139837860602501569, −1.24033268397176857425150351253, −0.65004160257863599080926783591, −0.42936451160579537924671702837, −0.27092515825783338549203393261,
0.27092515825783338549203393261, 0.42936451160579537924671702837, 0.65004160257863599080926783591, 1.24033268397176857425150351253, 1.56617192724139837860602501569, 1.84113175081442612585096897771, 2.03141536428772317914521004931, 2.06654955129862165083267573078, 2.19988385926848787495153851337, 2.27248337529066737729467379761, 2.85249728577045923299528853404, 2.94473902732425550788567762622, 3.26601454483731431835599146032, 3.73787013392988393287581521115, 3.74623448102970239794614645092, 3.78722775984697590495323101394, 4.11175821591785771036656512434, 4.19787940702584306602654924709, 4.27624149011911824441656667614, 4.30072384886374500052138437362, 4.80633183370004306612720727497, 4.87951498259642064143071842277, 5.16390021798114330520574733130, 5.41248176701585342978368169148, 5.54891211130720804574207155006
