| L(s) = 1 | + 2·3-s − 2·5-s − 6·7-s − 4·15-s − 4·19-s − 12·21-s − 6·23-s + 3·25-s − 2·27-s − 12·31-s + 12·35-s + 12·37-s − 12·41-s − 10·43-s − 6·47-s + 16·49-s − 8·57-s − 12·59-s − 12·61-s + 10·67-s − 12·69-s − 12·71-s + 8·73-s + 6·75-s − 24·79-s − 81-s + 6·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 2.26·7-s − 1.03·15-s − 0.917·19-s − 2.61·21-s − 1.25·23-s + 3/5·25-s − 0.384·27-s − 2.15·31-s + 2.02·35-s + 1.97·37-s − 1.87·41-s − 1.52·43-s − 0.875·47-s + 16/7·49-s − 1.05·57-s − 1.56·59-s − 1.53·61-s + 1.22·67-s − 1.44·69-s − 1.42·71-s + 0.936·73-s + 0.692·75-s − 2.70·79-s − 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.281360417978357017805351356161, −9.070234811663858553101705821719, −8.681001319532989974272797951141, −8.405035320388900658397097166454, −7.76558041424764154092022312267, −7.55257332695143277582702591170, −7.14954335477007668383248253911, −6.41377536925402254660438488987, −6.20094081640058115772318205601, −6.10423948468778849350470186281, −5.08331055880141259994210448130, −4.73717145696828253285189154289, −3.86601702308792731492741157077, −3.71724607066199712817685669252, −3.27651483520593012719071023419, −2.93033853627302621070556068692, −2.33456996454020072029998739069, −1.62671072304389720712087781313, 0, 0,
1.62671072304389720712087781313, 2.33456996454020072029998739069, 2.93033853627302621070556068692, 3.27651483520593012719071023419, 3.71724607066199712817685669252, 3.86601702308792731492741157077, 4.73717145696828253285189154289, 5.08331055880141259994210448130, 6.10423948468778849350470186281, 6.20094081640058115772318205601, 6.41377536925402254660438488987, 7.14954335477007668383248253911, 7.55257332695143277582702591170, 7.76558041424764154092022312267, 8.405035320388900658397097166454, 8.681001319532989974272797951141, 9.070234811663858553101705821719, 9.281360417978357017805351356161
