| L(s) = 1 | + 53·9-s − 78·11-s − 148·19-s + 474·29-s − 360·31-s − 696·41-s − 49·49-s − 904·59-s + 680·61-s + 1.05e3·71-s − 1.07e3·79-s + 2.08e3·81-s + 1.15e3·89-s − 4.13e3·99-s − 2.22e3·101-s − 2.64e3·109-s + 1.90e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.10e3·169-s + ⋯ |
| L(s) = 1 | + 1.96·9-s − 2.13·11-s − 1.78·19-s + 3.03·29-s − 2.08·31-s − 2.65·41-s − 1/7·49-s − 1.99·59-s + 1.42·61-s + 1.76·71-s − 1.53·79-s + 2.85·81-s + 1.37·89-s − 4.19·99-s − 2.19·101-s − 2.32·109-s + 1.42·121-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 + 1.86·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4318128016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4318128016\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 53 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 39 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4105 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9601 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 74 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24138 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 237 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 180 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 182 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 348 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158530 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 170397 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 254490 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 452 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 340 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 435062 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 528 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 471118 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 539 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1116678 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 576 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1141417 T^{2} + p^{6} T^{4} \) |
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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
−9.593868971895224206648029714899, −8.940944955963450639367474804702, −8.424054882522186447306253475699, −8.171413078634136908537524875465, −7.920529961195032972693815313484, −7.28841797703781597727048120502, −6.99156242643134609048778755363, −6.55746391108531440732822958011, −6.34752469178164307918408105169, −5.48519441501255379921576700627, −5.16342124125066597014129362318, −4.73085442306547874256104084151, −4.46502997166415703649771766725, −3.83316232941776231760179563424, −3.38742813321809926639082339710, −2.64872786152571821525163432820, −2.27446128636894474325582599336, −1.68005242519364892257185971992, −1.13819527193461986766429257193, −0.15188156673137321641448633039,
0.15188156673137321641448633039, 1.13819527193461986766429257193, 1.68005242519364892257185971992, 2.27446128636894474325582599336, 2.64872786152571821525163432820, 3.38742813321809926639082339710, 3.83316232941776231760179563424, 4.46502997166415703649771766725, 4.73085442306547874256104084151, 5.16342124125066597014129362318, 5.48519441501255379921576700627, 6.34752469178164307918408105169, 6.55746391108531440732822958011, 6.99156242643134609048778755363, 7.28841797703781597727048120502, 7.920529961195032972693815313484, 8.171413078634136908537524875465, 8.424054882522186447306253475699, 8.940944955963450639367474804702, 9.593868971895224206648029714899
