| L(s) = 1 | + 3·3-s − 5·7-s + 6·9-s + 7·13-s + 9·19-s − 15·21-s + 25-s + 9·27-s + 8·31-s + 3·37-s + 21·39-s + 10·43-s + 5·49-s + 27·57-s + 61-s − 30·63-s + 14·67-s − 27·73-s + 3·75-s − 21·79-s + 9·81-s − 35·91-s + 24·93-s − 6·97-s − 5·103-s + 15·109-s + 9·111-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.88·7-s + 2·9-s + 1.94·13-s + 2.06·19-s − 3.27·21-s + 1/5·25-s + 1.73·27-s + 1.43·31-s + 0.493·37-s + 3.36·39-s + 1.52·43-s + 5/7·49-s + 3.57·57-s + 0.128·61-s − 3.77·63-s + 1.71·67-s − 3.16·73-s + 0.346·75-s − 2.36·79-s + 81-s − 3.66·91-s + 2.48·93-s − 0.609·97-s − 0.492·103-s + 1.43·109-s + 0.854·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.196852761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.196852761\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 9 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.229299237409364275370762183434, −7.61189948442285257314175473863, −7.20252808828963951908110092463, −6.93044223276749010664864610539, −6.29399695392207029513377617445, −5.95147523226144292568940568494, −5.60529442721347893790710983793, −4.64658881466613405081567332387, −4.19293914216706563472881144061, −3.63172936159880860848084764231, −3.27462075633713565506735995658, −2.96900064682697732730241662382, −2.54339489221714071674098638496, −1.48004372233452398144902009884, −0.921868903154556544751501480945,
0.921868903154556544751501480945, 1.48004372233452398144902009884, 2.54339489221714071674098638496, 2.96900064682697732730241662382, 3.27462075633713565506735995658, 3.63172936159880860848084764231, 4.19293914216706563472881144061, 4.64658881466613405081567332387, 5.60529442721347893790710983793, 5.95147523226144292568940568494, 6.29399695392207029513377617445, 6.93044223276749010664864610539, 7.20252808828963951908110092463, 7.61189948442285257314175473863, 8.229299237409364275370762183434
