L(s) = 1 | + 4-s + 2·5-s + 16-s − 6·17-s + 2·20-s − 3·25-s − 10·37-s − 16·41-s − 10·43-s + 14·47-s + 16·59-s + 64-s + 4·67-s − 6·68-s − 12·79-s + 2·80-s − 12·83-s − 12·85-s − 16·89-s − 3·100-s + 4·101-s + 4·109-s − 13·121-s − 10·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 0.894·5-s + 1/4·16-s − 1.45·17-s + 0.447·20-s − 3/5·25-s − 1.64·37-s − 2.49·41-s − 1.52·43-s + 2.04·47-s + 2.08·59-s + 1/8·64-s + 0.488·67-s − 0.727·68-s − 1.35·79-s + 0.223·80-s − 1.31·83-s − 1.30·85-s − 1.69·89-s − 0.299·100-s + 0.398·101-s + 0.383·109-s − 1.18·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
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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
−8.245434911163522179942131896458, −7.40654818788603568123567819672, −6.99252573337364764107289304698, −6.78612548464342363545429212646, −6.35001468641052098851909960781, −5.64272634046843917739300992251, −5.48239853492490009131896414899, −4.92062288672442932083282428879, −4.27355941636326996832260805751, −3.74253949199601787379469790972, −3.16200492318022523532089864977, −2.41038112635067118788471646762, −1.99609970980440244853994942412, −1.43938001501524395652906031811, 0,
1.43938001501524395652906031811, 1.99609970980440244853994942412, 2.41038112635067118788471646762, 3.16200492318022523532089864977, 3.74253949199601787379469790972, 4.27355941636326996832260805751, 4.92062288672442932083282428879, 5.48239853492490009131896414899, 5.64272634046843917739300992251, 6.35001468641052098851909960781, 6.78612548464342363545429212646, 6.99252573337364764107289304698, 7.40654818788603568123567819672, 8.245434911163522179942131896458