| L(s) = 1 | − 4·4-s − 24·11-s + 16·16-s − 58·19-s + 204·29-s − 530·31-s − 480·41-s + 96·44-s + 325·49-s + 204·59-s − 206·61-s − 64·64-s + 1.16e3·71-s + 232·76-s − 346·79-s − 1.64e3·89-s + 2.40e3·101-s + 410·109-s − 816·116-s − 2.23e3·121-s + 2.12e3·124-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 − 0.700·19-s + 1.30·29-s − 3.07·31-s − 1.82·41-s + 0.328·44-s + 0.947·49-s + 0.450·59-s − 0.432·61-s − 1/8·64-s + 1.94·71-s + 0.350·76-s − 0.492·79-s − 1.95·89-s + 2.36·101-s + 0.360·109-s − 0.653·116-s − 1.67·121-s + 1.53·124-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
| \(L(2)\) |
\(\approx\) |
\(0.9814020221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9814020221\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 325 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1894 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6050 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 29 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24010 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 265 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 97081 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 240 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 24325 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 202462 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 p^{2} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 103 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 598822 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 582 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 773809 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 173 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 p^{2} T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 822 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1151305 T^{2} + p^{6} T^{4} \) |
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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
−9.514722640522516713720084080943, −9.044839400573014321739698315051, −8.505474910460636966954472647118, −8.355054199684811333600483762219, −7.956717033656025061940255123310, −7.35486010532014143655802021831, −6.87795405972347978049613971065, −6.85195754635980544980307388766, −5.88767423365072053540411895551, −5.79330610213734836554212385894, −5.14906739888749657760577930802, −4.93776530159258239479375766870, −4.26537758097905627221776236942, −3.92227835690607220194654097353, −3.29174067534105005539050364849, −2.97511197426508658936461608173, −1.97459121257433797691399735334, −1.95366534347111400946750710533, −0.918963624311058904636843251437, −0.26942923598692617693567142832,
0.26942923598692617693567142832, 0.918963624311058904636843251437, 1.95366534347111400946750710533, 1.97459121257433797691399735334, 2.97511197426508658936461608173, 3.29174067534105005539050364849, 3.92227835690607220194654097353, 4.26537758097905627221776236942, 4.93776530159258239479375766870, 5.14906739888749657760577930802, 5.79330610213734836554212385894, 5.88767423365072053540411895551, 6.85195754635980544980307388766, 6.87795405972347978049613971065, 7.35486010532014143655802021831, 7.956717033656025061940255123310, 8.355054199684811333600483762219, 8.505474910460636966954472647118, 9.044839400573014321739698315051, 9.514722640522516713720084080943
