| L(s) = 1 | − 4·4-s − 54·11-s + 16·16-s − 70·19-s − 240·29-s + 364·31-s − 714·41-s + 216·44-s − 470·49-s − 1.68e3·59-s − 476·61-s − 64·64-s + 1.41e3·71-s + 280·76-s − 1.30e3·79-s + 1.47e3·89-s − 924·101-s − 460·109-s + 960·116-s − 475·121-s − 1.45e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.48·11-s + 1/4·16-s − 0.845·19-s − 1.53·29-s + 2.10·31-s − 2.71·41-s + 0.740·44-s − 1.37·49-s − 3.70·59-s − 0.999·61-s − 1/8·64-s + 2.36·71-s + 0.422·76-s − 1.85·79-s + 1.75·89-s − 0.910·101-s − 0.404·109-s + 0.768·116-s − 0.356·121-s − 1.05·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1173667221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1173667221\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 470 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 27 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3610 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9385 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 35 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 18250 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 79990 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 357 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 137110 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 200590 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 195050 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 840 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 238 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 389005 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 708 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 760345 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 650 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 328165 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 735 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 602110 T^{2} + p^{6} T^{4} \) |
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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
−11.35568638200711736463865723103, −10.33780407002663614960805793318, −10.20386161817641087865381003514, −9.431014774144783312555474101921, −9.268240080196071559951718991828, −8.463509399496716743683138674638, −8.224290522001209330590792524291, −7.80019920591031056105901603379, −7.38667708460933914260209778934, −6.47252091606453736172641999980, −6.39927166337522203123580360053, −5.61630957668637738770397233042, −4.99029791363619424145833822362, −4.81157660181719653220600379830, −4.13000755981286065308972713479, −3.32552159689396697349887064991, −2.92843645948722397101641318163, −2.09063581339734631606879067464, −1.39375735391236763612693226048, −0.11070624202348647325579644946,
0.11070624202348647325579644946, 1.39375735391236763612693226048, 2.09063581339734631606879067464, 2.92843645948722397101641318163, 3.32552159689396697349887064991, 4.13000755981286065308972713479, 4.81157660181719653220600379830, 4.99029791363619424145833822362, 5.61630957668637738770397233042, 6.39927166337522203123580360053, 6.47252091606453736172641999980, 7.38667708460933914260209778934, 7.80019920591031056105901603379, 8.224290522001209330590792524291, 8.463509399496716743683138674638, 9.268240080196071559951718991828, 9.431014774144783312555474101921, 10.20386161817641087865381003514, 10.33780407002663614960805793318, 11.35568638200711736463865723103
