| L(s) = 1 | − 15·5-s + 8·7-s − 10·11-s + 48·13-s − 37·17-s − 29·19-s − 11·23-s + 150·25-s − 28·29-s − 41·31-s − 120·35-s + 230·37-s − 370·41-s + 130·43-s − 56·47-s − 209·49-s − 805·53-s + 150·55-s + 576·59-s − 257·61-s − 720·65-s + 14·67-s + 1.23e3·71-s − 398·73-s − 80·77-s + 321·79-s + 687·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.431·7-s − 0.274·11-s + 1.02·13-s − 0.527·17-s − 0.350·19-s − 0.0997·23-s + 6/5·25-s − 0.179·29-s − 0.237·31-s − 0.579·35-s + 1.02·37-s − 1.40·41-s + 0.461·43-s − 0.173·47-s − 0.609·49-s − 2.08·53-s + 0.367·55-s + 1.27·59-s − 0.539·61-s − 1.37·65-s + 0.0255·67-s + 2.06·71-s − 0.638·73-s − 0.118·77-s + 0.457·79-s + 0.908·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.056616453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.056616453\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 8 T + 39 p T^{2} - 11320 T^{3} + 39 p^{4} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 10 T + 1129 T^{2} - 2980 p T^{3} + 1129 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 48 T + 3891 T^{2} - 92328 T^{3} + 3891 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 37 T + 14418 T^{2} + 347965 T^{3} + 14418 p^{3} T^{4} + 37 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 29 T + 17960 T^{2} + 420497 T^{3} + 17960 p^{3} T^{4} + 29 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 11 T + 30456 T^{2} + 222899 T^{3} + 30456 p^{3} T^{4} + 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 28 T + 55099 T^{2} + 1662544 T^{3} + 55099 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 41 T + 78964 T^{2} + 2734453 T^{3} + 78964 p^{3} T^{4} + 41 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 230 T + 114787 T^{2} - 19298596 T^{3} + 114787 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 370 T + 214275 T^{2} + 48796108 T^{3} + 214275 p^{3} T^{4} + 370 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 130 T + 44673 T^{2} - 3936068 T^{3} + 44673 p^{3} T^{4} - 130 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 56 T + 115517 T^{2} + 4195536 T^{3} + 115517 p^{3} T^{4} + 56 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 805 T + 594702 T^{2} + 247415005 T^{3} + 594702 p^{3} T^{4} + 805 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 576 T + 3639 p T^{2} - 9306072 T^{3} + 3639 p^{4} T^{4} - 576 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 257 T + 202474 T^{2} + 153986701 T^{3} + 202474 p^{3} T^{4} + 257 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 14 T + 651149 T^{2} + 22117636 T^{3} + 651149 p^{3} T^{4} - 14 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1238 T + 1448677 T^{2} - 895305068 T^{3} + 1448677 p^{3} T^{4} - 1238 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 398 T + 1211307 T^{2} + 310732420 T^{3} + 1211307 p^{3} T^{4} + 398 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 321 T + 774636 T^{2} - 459949637 T^{3} + 774636 p^{3} T^{4} - 321 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 687 T + 1037148 T^{2} - 349696983 T^{3} + 1037148 p^{3} T^{4} - 687 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2358 T + 3793395 T^{2} + 3699365060 T^{3} + 3793395 p^{3} T^{4} + 2358 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 576 T + 2295459 T^{2} - 793944704 T^{3} + 2295459 p^{3} T^{4} - 576 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.898542612646149894491172648043, −7.48771721411008604669341244824, −7.20062389627834473503517601623, −6.91874678248098538665194116291, −6.54525637227768425840281942153, −6.50762781886380997598705549825, −6.27153002861706750575730209513, −5.77052802858411494790863932836, −5.48901015247237656875389974818, −5.28573482638398320642356962815, −4.94004841203514836853986461587, −4.62743202223805631419636825434, −4.40558758761880139655108536113, −4.01623971858255912500607427275, −3.87530596506929639928862302611, −3.66395363599345291434115072331, −3.02665822920982563008422407658, −2.92747259777681459064573381528, −2.80299950164098251830264048593, −1.93071564174774239903872652348, −1.72474942224237824769599321239, −1.68821790046771359615436816830, −0.825906194123945619465300611495, −0.60588601962814371971005132149, −0.33296650176582609211933709773,
0.33296650176582609211933709773, 0.60588601962814371971005132149, 0.825906194123945619465300611495, 1.68821790046771359615436816830, 1.72474942224237824769599321239, 1.93071564174774239903872652348, 2.80299950164098251830264048593, 2.92747259777681459064573381528, 3.02665822920982563008422407658, 3.66395363599345291434115072331, 3.87530596506929639928862302611, 4.01623971858255912500607427275, 4.40558758761880139655108536113, 4.62743202223805631419636825434, 4.94004841203514836853986461587, 5.28573482638398320642356962815, 5.48901015247237656875389974818, 5.77052802858411494790863932836, 6.27153002861706750575730209513, 6.50762781886380997598705549825, 6.54525637227768425840281942153, 6.91874678248098538665194116291, 7.20062389627834473503517601623, 7.48771721411008604669341244824, 7.898542612646149894491172648043
