| L(s) = 1 | + 4·4-s − 50·25-s − 144·29-s − 72·41-s + 20·49-s − 296·61-s − 64·64-s − 72·89-s − 200·100-s − 144·101-s + 104·109-s − 576·116-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 288·164-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 4-s − 2·25-s − 4.96·29-s − 1.75·41-s + 0.408·49-s − 4.85·61-s − 64-s − 0.808·89-s − 2·100-s − 1.42·101-s + 0.954·109-s − 4.96·116-s + 2.21·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s − 1.75·164-s + 0.00598·167-s + 0.165·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6778634625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6778634625\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 36 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 1874 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 178 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 4942 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5990 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 7250 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 718 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 9362 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 4370 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 5666 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 13634 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.153951718552026362492184599664, −9.043730863199164944235358007324, −8.374771765451133634202781992725, −7.980647666964529344070320259314, −7.951778713441264346579436727135, −7.73417020219251135542917084716, −7.32608154982488439577829728401, −7.08013393255983853849587827735, −6.91695597816170132434497235598, −6.67821230283248354552895136562, −5.96799338933057284703192938681, −5.86733273706036309341186507352, −5.68791583125425206057988374036, −5.65597396592379580872745693064, −4.89679929304748720896652796879, −4.67170285236554176737861702631, −4.12165445256262683031701901796, −3.99394460265033085792277533667, −3.43742046502072454336179897635, −3.20545555520036530562574252121, −2.82795598849732284461674779715, −1.84468990945888461962682378871, −1.83729652023832807816969641358, −1.80905440824910575948112813922, −0.23041075168654342416447235850,
0.23041075168654342416447235850, 1.80905440824910575948112813922, 1.83729652023832807816969641358, 1.84468990945888461962682378871, 2.82795598849732284461674779715, 3.20545555520036530562574252121, 3.43742046502072454336179897635, 3.99394460265033085792277533667, 4.12165445256262683031701901796, 4.67170285236554176737861702631, 4.89679929304748720896652796879, 5.65597396592379580872745693064, 5.68791583125425206057988374036, 5.86733273706036309341186507352, 5.96799338933057284703192938681, 6.67821230283248354552895136562, 6.91695597816170132434497235598, 7.08013393255983853849587827735, 7.32608154982488439577829728401, 7.73417020219251135542917084716, 7.951778713441264346579436727135, 7.980647666964529344070320259314, 8.374771765451133634202781992725, 9.043730863199164944235358007324, 9.153951718552026362492184599664
