| L(s) = 1 | − 6·2-s − 9·3-s + 24·4-s − 6·5-s + 54·6-s + 21·7-s − 80·8-s + 54·9-s + 36·10-s − 15·11-s − 216·12-s + 39·13-s − 126·14-s + 54·15-s + 240·16-s + 9·17-s − 324·18-s + 60·19-s − 144·20-s − 189·21-s + 90·22-s − 162·23-s + 720·24-s − 150·25-s − 234·26-s − 270·27-s + 504·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s − 0.536·5-s + 3.67·6-s + 1.13·7-s − 3.53·8-s + 2·9-s + 1.13·10-s − 0.411·11-s − 5.19·12-s + 0.832·13-s − 2.40·14-s + 0.929·15-s + 15/4·16-s + 0.128·17-s − 4.24·18-s + 0.724·19-s − 1.60·20-s − 1.96·21-s + 0.872·22-s − 1.46·23-s + 6.12·24-s − 6/5·25-s − 1.76·26-s − 1.92·27-s + 3.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7936274003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7936274003\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 6 T + 186 T^{2} + 474 T^{3} + 186 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 15 T + 15 p^{2} T^{2} + 59154 T^{3} + 15 p^{5} T^{4} + 15 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 9 T + 9069 T^{2} + 92358 T^{3} + 9069 p^{3} T^{4} - 9 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 60 T + 5496 T^{2} - 50852 p T^{3} + 5496 p^{3} T^{4} - 60 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 162 T + 30642 T^{2} + 3899772 T^{3} + 30642 p^{3} T^{4} + 162 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 138 T + 43494 T^{2} + 7459122 T^{3} + 43494 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 105 T + 58029 T^{2} - 5233646 T^{3} + 58029 p^{3} T^{4} - 105 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 309 T + 136617 T^{2} + 29017690 T^{3} + 136617 p^{3} T^{4} + 309 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 408 T + 253239 T^{2} + 57772272 T^{3} + 253239 p^{3} T^{4} + 408 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 330 T + 143358 T^{2} - 55247096 T^{3} + 143358 p^{3} T^{4} - 330 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 69 T + 122397 T^{2} + 11991318 T^{3} + 122397 p^{3} T^{4} + 69 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 57 T + 257055 T^{2} + 11319582 T^{3} + 257055 p^{3} T^{4} + 57 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 36 T + 32937 T^{2} - 105666264 T^{3} + 32937 p^{3} T^{4} - 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1275 T + 1075479 T^{2} - 603951626 T^{3} + 1075479 p^{3} T^{4} - 1275 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1392 T + 1272021 T^{2} - 782607440 T^{3} + 1272021 p^{3} T^{4} - 1392 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1416 T + 1493061 T^{2} - 1004258928 T^{3} + 1493061 p^{3} T^{4} - 1416 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 744 T + 986388 T^{2} - 577169138 T^{3} + 986388 p^{3} T^{4} - 744 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1275 T + 1514823 T^{2} - 1001594234 T^{3} + 1514823 p^{3} T^{4} - 1275 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 45 T + 1269699 T^{2} - 63102582 T^{3} + 1269699 p^{3} T^{4} - 45 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 189 T + 1187295 T^{2} + 134882874 T^{3} + 1187295 p^{3} T^{4} - 189 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2409 T + 3946767 T^{2} - 4325156750 T^{3} + 3946767 p^{3} T^{4} - 2409 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.439642101131864647517908122210, −8.822287200413303689292966992155, −8.495383539084339892197453012517, −8.367548252576446672912104674149, −7.87099686734744251636061599326, −7.86896969397478682937142028017, −7.65938230431082543888011459419, −6.97287490053049134230941548866, −6.81412134708706035088197729227, −6.80048840754731137577740009633, −6.10223551042681472035087004775, −5.75910150469445016295521621660, −5.73339830702828850541469985540, −5.19331158430705620086763748570, −4.95435174404077631827890993565, −4.50418831563767949085190291090, −3.84094927669916456093984937943, −3.52995530558812630940893789975, −3.43323718850180121644998714181, −2.10923146846846687555199773621, −2.03429031494855937010414066743, −1.92152869405691744546905738395, −0.973676120051190445376242087771, −0.58273012717145813111887711978, −0.54372924114894329215118589321,
0.54372924114894329215118589321, 0.58273012717145813111887711978, 0.973676120051190445376242087771, 1.92152869405691744546905738395, 2.03429031494855937010414066743, 2.10923146846846687555199773621, 3.43323718850180121644998714181, 3.52995530558812630940893789975, 3.84094927669916456093984937943, 4.50418831563767949085190291090, 4.95435174404077631827890993565, 5.19331158430705620086763748570, 5.73339830702828850541469985540, 5.75910150469445016295521621660, 6.10223551042681472035087004775, 6.80048840754731137577740009633, 6.81412134708706035088197729227, 6.97287490053049134230941548866, 7.65938230431082543888011459419, 7.86896969397478682937142028017, 7.87099686734744251636061599326, 8.367548252576446672912104674149, 8.495383539084339892197453012517, 8.822287200413303689292966992155, 9.439642101131864647517908122210
