| L(s) = 1 | + 6·2-s + 19·4-s + 36·8-s + 21·16-s + 36·17-s + 32·19-s + 24·23-s − 2·31-s − 126·32-s + 216·34-s + 192·38-s + 144·46-s − 12·47-s + 73·49-s + 54·53-s − 152·61-s − 12·62-s − 553·64-s + 684·68-s + 608·76-s − 28·79-s − 6·83-s + 456·92-s − 72·94-s + 438·98-s + 324·106-s + 378·107-s + ⋯ |
| L(s) = 1 | + 3·2-s + 19/4·4-s + 9/2·8-s + 1.31·16-s + 2.11·17-s + 1.68·19-s + 1.04·23-s − 0.0645·31-s − 3.93·32-s + 6.35·34-s + 5.05·38-s + 3.13·46-s − 0.255·47-s + 1.48·49-s + 1.01·53-s − 2.49·61-s − 0.193·62-s − 8.64·64-s + 10.0·68-s + 8·76-s − 0.354·79-s − 0.0722·83-s + 4.95·92-s − 0.765·94-s + 4.46·98-s + 3.05·106-s + 3.53·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(15.55372323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.55372323\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 782 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2338 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 238 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 1198 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 27 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6062 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8878 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1982 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6433 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7742 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 11593 T^{2} + p^{4} T^{4} \) |
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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
−10.64851599175318632055552668802, −10.42240909120571122998171852432, −9.582167197817375109862810087086, −9.388483494944059920257428679423, −8.842134294887871721329781376839, −8.155070555436612444889929464082, −7.40557003774436145508132022684, −7.34969937871699660532418298244, −6.76698480526142161823728617350, −6.14202149878752256502868416011, −5.59944052124229439036062918436, −5.57835892865768253019660573957, −5.04207731762770420139637553484, −4.58784017769784777864863934685, −4.01892325873428724013848435848, −3.49634623835504241710561409842, −2.98001924255302975424697300431, −2.92481409051240291336276720271, −1.84198216154308183204170517647, −0.894699824673261467153978948570,
0.894699824673261467153978948570, 1.84198216154308183204170517647, 2.92481409051240291336276720271, 2.98001924255302975424697300431, 3.49634623835504241710561409842, 4.01892325873428724013848435848, 4.58784017769784777864863934685, 5.04207731762770420139637553484, 5.57835892865768253019660573957, 5.59944052124229439036062918436, 6.14202149878752256502868416011, 6.76698480526142161823728617350, 7.34969937871699660532418298244, 7.40557003774436145508132022684, 8.155070555436612444889929464082, 8.842134294887871721329781376839, 9.388483494944059920257428679423, 9.582167197817375109862810087086, 10.42240909120571122998171852432, 10.64851599175318632055552668802
