| L(s) = 1 | + 4·2-s + 12·4-s + 32·8-s + 18·9-s + 80·16-s + 72·18-s − 48·25-s − 80·29-s + 192·32-s + 216·36-s − 48·37-s − 192·50-s − 180·53-s − 320·58-s + 448·64-s + 576·72-s − 192·74-s + 243·81-s − 576·100-s − 720·106-s + 240·109-s + 60·113-s − 960·116-s + 242·121-s + 127-s + 1.02e3·128-s + 131-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 2·9-s + 5·16-s + 4·18-s − 1.91·25-s − 2.75·29-s + 6·32-s + 6·36-s − 1.29·37-s − 3.83·50-s − 3.39·53-s − 5.51·58-s + 7·64-s + 8·72-s − 2.59·74-s + 3·81-s − 5.75·100-s − 6.79·106-s + 2.20·109-s + 0.530·113-s − 8.27·116-s + 2·121-s + 0.00787·127-s + 8·128-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.468897605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.468897605\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 48 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 240 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 480 T^{2} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 1440 T^{2} + p^{4} T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 90 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 2640 T^{2} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 10560 T^{2} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12480 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18720 T^{2} + p^{4} T^{4} \) |
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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
−12.50253558320000205952189427732, −12.39654354452379733516005459226, −11.66190909579143122276434895134, −11.20082057830641140824196950599, −10.84251826046443850059361702498, −10.19160565300150037263457094531, −9.754399874224431765245108184445, −9.317593332496501984542966825463, −7.999456689505247356627254791528, −7.73080274000703704474193975056, −7.12699510349989248123107152531, −6.85764733959052772624458418871, −5.97508869433613277027620250328, −5.70540597267753828594049404984, −4.78490339785699574423294206504, −4.46897164182764683599057966275, −3.65858846889893197942071872472, −3.43581985813052847828662029175, −1.86967472601457373171388787514, −1.79550254369905278159745837884,
1.79550254369905278159745837884, 1.86967472601457373171388787514, 3.43581985813052847828662029175, 3.65858846889893197942071872472, 4.46897164182764683599057966275, 4.78490339785699574423294206504, 5.70540597267753828594049404984, 5.97508869433613277027620250328, 6.85764733959052772624458418871, 7.12699510349989248123107152531, 7.73080274000703704474193975056, 7.999456689505247356627254791528, 9.317593332496501984542966825463, 9.754399874224431765245108184445, 10.19160565300150037263457094531, 10.84251826046443850059361702498, 11.20082057830641140824196950599, 11.66190909579143122276434895134, 12.39654354452379733516005459226, 12.50253558320000205952189427732
