L(s) = 1 | + 3-s − 3·4-s − 4·5-s + 9-s − 3·12-s − 4·15-s + 5·16-s − 12·17-s + 12·20-s + 2·25-s + 27-s − 3·36-s − 20·37-s − 4·41-s − 8·43-s − 4·45-s + 24·47-s + 5·48-s − 7·49-s − 12·51-s − 24·59-s + 12·60-s − 3·64-s − 8·67-s + 36·68-s + 2·75-s − 20·80-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 3/2·4-s − 1.78·5-s + 1/3·9-s − 0.866·12-s − 1.03·15-s + 5/4·16-s − 2.91·17-s + 2.68·20-s + 2/5·25-s + 0.192·27-s − 1/2·36-s − 3.28·37-s − 0.624·41-s − 1.21·43-s − 0.596·45-s + 3.50·47-s + 0.721·48-s − 49-s − 1.68·51-s − 3.12·59-s + 1.54·60-s − 3/8·64-s − 0.977·67-s + 4.36·68-s + 0.230·75-s − 2.23·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.263513432862467471567669978548, −7.75382334304440982742558846703, −7.34752150402139467995584348966, −6.78418502492955034999702547464, −6.46787589567348650226556085189, −5.56436473344957075485364042125, −4.94046317313085415521526307990, −4.56991321080597356703354311172, −4.20789833671809510725213788268, −3.70760760284754853820305649861, −3.44365518362789223021999993571, −2.47817878081925227300773446057, −1.65980873553023464631625543363, 0, 0,
1.65980873553023464631625543363, 2.47817878081925227300773446057, 3.44365518362789223021999993571, 3.70760760284754853820305649861, 4.20789833671809510725213788268, 4.56991321080597356703354311172, 4.94046317313085415521526307990, 5.56436473344957075485364042125, 6.46787589567348650226556085189, 6.78418502492955034999702547464, 7.34752150402139467995584348966, 7.75382334304440982742558846703, 8.263513432862467471567669978548