| L(s) = 1 | − 4·4-s − 46·9-s + 18·11-s + 16·16-s + 20·19-s − 378·29-s − 464·31-s + 184·36-s − 876·41-s − 72·44-s − 49·49-s + 1.34e3·59-s + 412·61-s − 64·64-s − 942·71-s − 80·76-s − 1.48e3·79-s + 1.38e3·81-s − 360·89-s − 828·99-s − 1.45e3·101-s + 1.38e3·109-s + 1.51e3·116-s − 2.41e3·121-s + 1.85e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.70·9-s + 0.493·11-s + 1/4·16-s + 0.241·19-s − 2.42·29-s − 2.68·31-s + 0.851·36-s − 3.33·41-s − 0.246·44-s − 1/7·49-s + 2.96·59-s + 0.864·61-s − 1/8·64-s − 1.57·71-s − 0.120·76-s − 2.11·79-s + 1.90·81-s − 0.428·89-s − 0.840·99-s − 1.43·101-s + 1.21·109-s + 1.21·116-s − 1.81·121-s + 1.34·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09424591143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09424591143\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 46 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 9 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 p T + p^{3} T^{2} )( 1 + 6 p T + p^{3} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 610 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 18709 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 189 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 232 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 8281 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 438 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 34405 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 28550 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 172438 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 672 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 206 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 242725 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 471 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 401038 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 743 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 p^{2} T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 180 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1791490 T^{2} + p^{6} T^{4} \) |
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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
−11.66192988503986676094214094391, −10.85739241904669680892210353223, −10.44394703519454479390961041763, −9.769174214227922185839992031918, −9.397529289028950408330502653531, −8.932920847735023648826039712272, −8.455863350980625372804025968261, −8.333809676569906029628408376742, −7.26627884511919658844955868587, −7.25307312667974695041238543022, −6.44710302245667124470664540980, −5.67259986714009165677218911052, −5.44139187315020638334192370373, −5.14252532814717964385580122860, −4.01036463542792585896394670172, −3.65092981487146002651057922090, −3.13735781912030899113490882638, −2.17553701446599698400587380651, −1.51419165046458572686692626278, −0.10630053046025347886711834822,
0.10630053046025347886711834822, 1.51419165046458572686692626278, 2.17553701446599698400587380651, 3.13735781912030899113490882638, 3.65092981487146002651057922090, 4.01036463542792585896394670172, 5.14252532814717964385580122860, 5.44139187315020638334192370373, 5.67259986714009165677218911052, 6.44710302245667124470664540980, 7.25307312667974695041238543022, 7.26627884511919658844955868587, 8.333809676569906029628408376742, 8.455863350980625372804025968261, 8.932920847735023648826039712272, 9.397529289028950408330502653531, 9.769174214227922185839992031918, 10.44394703519454479390961041763, 10.85739241904669680892210353223, 11.66192988503986676094214094391
