| L(s) = 1 | − 2·4-s − 28·7-s − 36·19-s − 50·25-s + 56·28-s − 6·31-s + 106·37-s − 116·43-s + 490·49-s − 30·61-s + 8·64-s − 10·67-s + 180·73-s + 72·76-s + 50·79-s + 100·100-s − 150·103-s − 398·109-s + 224·121-s + 12·124-s + 127-s + 131-s + 1.00e3·133-s + 137-s + 139-s − 212·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 4·7-s − 1.89·19-s − 2·25-s + 2·28-s − 0.193·31-s + 2.86·37-s − 2.69·43-s + 10·49-s − 0.491·61-s + 1/8·64-s − 0.149·67-s + 2.46·73-s + 0.947·76-s + 0.632·79-s + 100-s − 1.45·103-s − 3.65·109-s + 1.85·121-s + 3/31·124-s + 0.00787·127-s + 0.00763·131-s + 7.57·133-s + 0.00729·137-s + 0.00719·139-s − 1.43·148-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04539075161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04539075161\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 224 T^{2} + 35535 T^{4} - 224 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 335 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 92 T^{2} - 75057 T^{4} + 92 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 18 T + 469 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 400 T^{2} - 119841 T^{4} + 400 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 1520 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 3 T + 964 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 53 T + 1440 T^{2} - 53 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2012 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 29 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 4364 T^{2} + 14164815 T^{4} + 4364 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 2576 T^{2} - 1254705 T^{4} - 2576 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 428 T^{2} - 11934177 T^{4} + 428 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 15 T + 3796 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 5 T - 4464 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 8930 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 90 T + 8029 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 25 T - 5616 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 5716 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 5258 T^{2} - 35095677 T^{4} + 5258 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 17735 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.84360566368666285838279594158, −7.84228076793031043183176136562, −7.63785366043044870266094712765, −7.16167354918983854341960317624, −6.77671124087357778505605828419, −6.51312142479901605526771280626, −6.49144685615567376144891110314, −6.44559312627242889723523418414, −6.15669105034623406094611430620, −5.76980166110393108346409466640, −5.42520384538231127824106791495, −5.38764254419681485706762497438, −4.85993177522402401346987401875, −4.34141088249966857130273841167, −4.11349417029396705247847465778, −3.84021206862539605140731765270, −3.81765964637267939735094036723, −3.36962899933375661382865957098, −3.02649857749829985515718178481, −2.64532964247510023364775356484, −2.45325465518662171710011370679, −2.07954386235405384003385899218, −1.30920666302628898974747489908, −0.55249634720876751303182239377, −0.07846637302079533758813522699,
0.07846637302079533758813522699, 0.55249634720876751303182239377, 1.30920666302628898974747489908, 2.07954386235405384003385899218, 2.45325465518662171710011370679, 2.64532964247510023364775356484, 3.02649857749829985515718178481, 3.36962899933375661382865957098, 3.81765964637267939735094036723, 3.84021206862539605140731765270, 4.11349417029396705247847465778, 4.34141088249966857130273841167, 4.85993177522402401346987401875, 5.38764254419681485706762497438, 5.42520384538231127824106791495, 5.76980166110393108346409466640, 6.15669105034623406094611430620, 6.44559312627242889723523418414, 6.49144685615567376144891110314, 6.51312142479901605526771280626, 6.77671124087357778505605828419, 7.16167354918983854341960317624, 7.63785366043044870266094712765, 7.84228076793031043183176136562, 7.84360566368666285838279594158
