| L(s) = 1 | − 4·4-s − 24·11-s + 16·16-s + 314·19-s − 132·29-s − 50·31-s − 1.00e3·41-s + 96·44-s + 202·49-s + 636·59-s + 586·61-s − 64·64-s − 240·71-s − 1.25e3·76-s − 1.83e3·79-s − 2.54e3·89-s − 984·101-s + 1.81e3·109-s + 528·116-s − 2.23e3·121-s + 200·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.657·11-s + 1/4·16-s + 3.79·19-s − 0.845·29-s − 0.289·31-s − 3.83·41-s + 0.328·44-s + 0.588·49-s + 1.40·59-s + 1.22·61-s − 1/8·64-s − 0.401·71-s − 1.89·76-s − 2.61·79-s − 3.02·89-s − 0.969·101-s + 1.59·109-s + 0.422·116-s − 1.67·121-s + 0.144·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9654331999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9654331999\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 202 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1199 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 157 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10645 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 66 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 25 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2710 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 504 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 14614 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 144142 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 297745 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 318 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 293 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 497842 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 776098 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 917 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1048093 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1272 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 61762 T^{2} + p^{6} T^{4} \) |
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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
−9.547309246470279361367649356665, −8.908779307400659211550841679580, −8.774161673056846870998681420824, −8.181707610513285595457663948353, −7.86477713327128268962068854360, −7.34644571316452324760322061060, −7.07933726437958444918465428674, −6.80127324522516639880146112948, −5.96418774353224261746537223386, −5.35507128140237101930150216017, −5.33029618947511257544340545158, −5.13176057874360848695290270392, −4.28655536222325652550279685306, −3.76935632079978045677505634475, −3.25996529209873327360309095430, −3.03730823304192081714980406823, −2.31815909939069258153273437822, −1.38319702431786030342843126942, −1.22315563001457299801655859828, −0.24331252468986436478893736897,
0.24331252468986436478893736897, 1.22315563001457299801655859828, 1.38319702431786030342843126942, 2.31815909939069258153273437822, 3.03730823304192081714980406823, 3.25996529209873327360309095430, 3.76935632079978045677505634475, 4.28655536222325652550279685306, 5.13176057874360848695290270392, 5.33029618947511257544340545158, 5.35507128140237101930150216017, 5.96418774353224261746537223386, 6.80127324522516639880146112948, 7.07933726437958444918465428674, 7.34644571316452324760322061060, 7.86477713327128268962068854360, 8.181707610513285595457663948353, 8.774161673056846870998681420824, 8.908779307400659211550841679580, 9.547309246470279361367649356665
