L(s) = 1 | + 2·5-s + 9-s + 8·13-s + 4·17-s + 3·25-s − 4·29-s − 12·37-s + 12·41-s + 2·45-s + 49-s + 16·53-s − 20·61-s + 16·65-s + 8·73-s + 81-s + 8·85-s − 28·89-s + 16·97-s − 12·101-s + 4·109-s + 24·113-s + 8·117-s − 18·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1/3·9-s + 2.21·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s − 1.97·37-s + 1.87·41-s + 0.298·45-s + 1/7·49-s + 2.19·53-s − 2.56·61-s + 1.98·65-s + 0.936·73-s + 1/9·81-s + 0.867·85-s − 2.96·89-s + 1.62·97-s − 1.19·101-s + 0.383·109-s + 2.25·113-s + 0.739·117-s − 1.63·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.029590026\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.029590026\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.481553612925985512669604062490, −7.81835639585916182571375219056, −7.48284568639593696898416392075, −6.87259354452381471297775882823, −6.51502067396389468143134947504, −5.94033866886860292935603183313, −5.55375053090780051552954501584, −5.45803650924575687262352989502, −4.46195444778293602014826196668, −4.08706078179886277200739673348, −3.43992207551315519554345327677, −3.08927995906889680200761478861, −2.15715671918253819106635126062, −1.54792096213768469866947757162, −0.959800325238339118056269866099,
0.959800325238339118056269866099, 1.54792096213768469866947757162, 2.15715671918253819106635126062, 3.08927995906889680200761478861, 3.43992207551315519554345327677, 4.08706078179886277200739673348, 4.46195444778293602014826196668, 5.45803650924575687262352989502, 5.55375053090780051552954501584, 5.94033866886860292935603183313, 6.51502067396389468143134947504, 6.87259354452381471297775882823, 7.48284568639593696898416392075, 7.81835639585916182571375219056, 8.481553612925985512669604062490