| L(s) = 1 | − 4·2-s + 12·4-s − 32·8-s − 40·11-s + 80·16-s + 160·22-s − 96·23-s − 162·25-s + 332·29-s − 192·32-s − 156·37-s + 872·43-s − 480·44-s + 384·46-s + 648·50-s − 124·53-s − 1.32e3·58-s + 448·64-s + 1.16e3·67-s + 1.08e3·71-s + 624·74-s − 1.36e3·79-s − 3.48e3·86-s + 1.28e3·88-s − 1.15e3·92-s − 1.94e3·100-s + 496·106-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1.09·11-s + 5/4·16-s + 1.55·22-s − 0.870·23-s − 1.29·25-s + 2.12·29-s − 1.06·32-s − 0.693·37-s + 3.09·43-s − 1.64·44-s + 1.23·46-s + 1.83·50-s − 0.321·53-s − 3.00·58-s + 7/8·64-s + 2.11·67-s + 1.81·71-s + 0.980·74-s − 1.93·79-s − 4.37·86-s + 1.55·88-s − 1.30·92-s − 1.94·100-s + 0.454·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.396477489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396477489\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 162 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 82 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6658 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 13630 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 166 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 16990 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 78 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 17390 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 436 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 165054 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 32850 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 379954 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 580 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 544 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 417586 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 680 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1104766 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 842862 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1394146 T^{2} + p^{6} T^{4} \) |
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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
−9.881944254086725109713867452478, −9.739796767729301659215511325273, −9.078288219211756099877720822629, −8.645554856818648397240544249760, −8.286108012076047630829941006302, −7.991812670032227911841190541722, −7.41463614506399791913433137982, −7.33497925738591060805815500058, −6.50336696420769706392207862451, −6.30815947298552293840304675977, −5.54559894331370388570873416265, −5.49013918813724257453180789283, −4.50044143373944351641085518650, −4.19094003270597575466085943541, −3.30616827862340583578236505598, −2.87722347891372389917982398890, −2.12299408355174890962522496910, −1.95778499536803532488249251807, −0.73343367140495820008026629904, −0.57944176600634645086332289842,
0.57944176600634645086332289842, 0.73343367140495820008026629904, 1.95778499536803532488249251807, 2.12299408355174890962522496910, 2.87722347891372389917982398890, 3.30616827862340583578236505598, 4.19094003270597575466085943541, 4.50044143373944351641085518650, 5.49013918813724257453180789283, 5.54559894331370388570873416265, 6.30815947298552293840304675977, 6.50336696420769706392207862451, 7.33497925738591060805815500058, 7.41463614506399791913433137982, 7.991812670032227911841190541722, 8.286108012076047630829941006302, 8.645554856818648397240544249760, 9.078288219211756099877720822629, 9.739796767729301659215511325273, 9.881944254086725109713867452478
