| L(s) = 1 | − 11·4-s + 64·16-s − 328·19-s − 250·25-s + 464·31-s + 1.37e3·49-s + 716·61-s − 781·64-s + 3.60e3·76-s + 608·79-s + 2.75e3·100-s + 3.66e3·109-s − 5.32e3·121-s − 5.10e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.37·4-s + 16-s − 3.96·19-s − 2·25-s + 2.68·31-s + 4·49-s + 1.50·61-s − 1.52·64-s + 5.44·76-s + 0.865·79-s + 11/4·100-s + 3.22·109-s − 4·121-s − 3.69·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.224327882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224327882\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 + 11 T^{2} + 57 T^{4} + 11 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 9394 T^{2} + 64109667 T^{4} - 9394 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 164 T + 20037 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 14654 T^{2} + 66703827 T^{4} + 14654 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 232 T + 24033 T^{2} - 232 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 90034 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 88666 T^{2} - 14302701573 T^{4} - 88666 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 358 T - 98817 T^{2} - 358 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 304 T - 400623 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 469546 T^{2} - 106466927253 T^{4} - 469546 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
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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
−9.065283399789086825112730065768, −8.999867064618836355856340113530, −8.657837297765065797154187145747, −8.463468560511233810606929903383, −8.291996963272231124834197752744, −7.79561661449116956028900453729, −7.75101257544140288060404244865, −7.31586172370153884353975113018, −6.76715385890649328285736993788, −6.58637689037998824063191942587, −6.20678664300172587326299453825, −6.06058496397866354045731673376, −5.73598349090904568142740630507, −5.11089550279912590835415889996, −5.09435691653473785100693386155, −4.35202171592028220192543487350, −4.19952075609584456844054806808, −4.12094997205469458270856449969, −3.86645918455041812306170305149, −3.08947741491031091712916918956, −2.48041639092386491450575667598, −2.25995373726390120186956476055, −1.74318480958483440792629824469, −0.72383537371945490398221170596, −0.40109473387365059637802486695,
0.40109473387365059637802486695, 0.72383537371945490398221170596, 1.74318480958483440792629824469, 2.25995373726390120186956476055, 2.48041639092386491450575667598, 3.08947741491031091712916918956, 3.86645918455041812306170305149, 4.12094997205469458270856449969, 4.19952075609584456844054806808, 4.35202171592028220192543487350, 5.09435691653473785100693386155, 5.11089550279912590835415889996, 5.73598349090904568142740630507, 6.06058496397866354045731673376, 6.20678664300172587326299453825, 6.58637689037998824063191942587, 6.76715385890649328285736993788, 7.31586172370153884353975113018, 7.75101257544140288060404244865, 7.79561661449116956028900453729, 8.291996963272231124834197752744, 8.463468560511233810606929903383, 8.657837297765065797154187145747, 8.999867064618836355856340113530, 9.065283399789086825112730065768
