| L(s) = 1 | − 9·4-s − 100·11-s + 17·16-s + 88·19-s − 100·29-s + 216·31-s − 800·41-s + 900·44-s − 214·49-s + 100·59-s − 1.03e3·61-s + 423·64-s − 1.40e3·71-s − 792·76-s + 1.03e3·79-s − 3.00e3·89-s + 900·101-s − 3.50e3·109-s + 900·116-s + 4.83e3·121-s − 1.94e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 9/8·4-s − 2.74·11-s + 0.265·16-s + 1.06·19-s − 0.640·29-s + 1.25·31-s − 3.04·41-s + 3.08·44-s − 0.623·49-s + 0.220·59-s − 2.17·61-s + 0.826·64-s − 2.34·71-s − 1.19·76-s + 1.46·79-s − 3.57·89-s + 0.886·101-s − 3.08·109-s + 0.720·116-s + 3.63·121-s − 1.40·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1924707392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1924707392\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 9 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 214 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 50 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9726 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 50 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 99706 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 400 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 80614 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 129246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74346 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 50 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 518 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 569126 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 700 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 609934 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 516 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 707974 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1500 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 831554 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.87457439711213653144573917720, −11.90226641563819880895941908040, −10.93664102146650739864570834724, −10.34590466642390902687245017594, −10.26342067392734688634043475992, −9.607151681520150666607915739687, −9.188536664044603131129819466116, −8.391102290415745716946555274225, −8.215220408083746372622230023940, −7.67964093513980609663355485575, −7.15513680949114607785533508478, −6.39287484097062946556978106773, −5.52611357198371830376436419741, −5.18304052624005759087087821214, −4.84522119965145592496214452255, −4.09600498155342483155682809070, −3.10096222915132285453537143373, −2.75999104218547414904300907632, −1.56288019235294930346476013692, −0.18283141867728760409124942591,
0.18283141867728760409124942591, 1.56288019235294930346476013692, 2.75999104218547414904300907632, 3.10096222915132285453537143373, 4.09600498155342483155682809070, 4.84522119965145592496214452255, 5.18304052624005759087087821214, 5.52611357198371830376436419741, 6.39287484097062946556978106773, 7.15513680949114607785533508478, 7.67964093513980609663355485575, 8.215220408083746372622230023940, 8.391102290415745716946555274225, 9.188536664044603131129819466116, 9.607151681520150666607915739687, 10.26342067392734688634043475992, 10.34590466642390902687245017594, 10.93664102146650739864570834724, 11.90226641563819880895941908040, 12.87457439711213653144573917720
