| L(s) = 1 | − 2·3-s − 4·5-s + 2·9-s − 4·11-s + 8·15-s − 4·17-s + 6·19-s + 8·25-s − 6·27-s + 14·29-s + 12·31-s + 8·33-s + 6·37-s − 12·43-s − 8·45-s − 20·47-s − 49-s + 8·51-s + 10·53-s + 16·55-s − 12·57-s − 14·59-s + 4·61-s − 16·67-s − 16·75-s + 8·79-s + 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 2/3·9-s − 1.20·11-s + 2.06·15-s − 0.970·17-s + 1.37·19-s + 8/5·25-s − 1.15·27-s + 2.59·29-s + 2.15·31-s + 1.39·33-s + 0.986·37-s − 1.82·43-s − 1.19·45-s − 2.91·47-s − 1/7·49-s + 1.12·51-s + 1.37·53-s + 2.15·55-s − 1.58·57-s − 1.82·59-s + 0.512·61-s − 1.95·67-s − 1.84·75-s + 0.900·79-s + 11/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.145520862771066887239374850421, −8.011448880203797533884299140563, −7.61911714559719413092592010498, −7.45285611450903001395411246290, −6.62336589275315757970408163728, −6.60504228364565900026710850533, −6.34434427454256381569107007670, −5.69442957324611895193295124170, −5.20984902525311716493396149347, −4.85158628959594320109486251729, −4.55845557930761765672675216511, −4.45743586498327452295321344360, −3.62707727887976064415127950913, −3.34932312375998979076903337646, −2.72124377519638254412480312830, −2.57356930524888605710413905810, −1.40816621546170426044088423240, −0.997819047486142157950637454039, 0, 0,
0.997819047486142157950637454039, 1.40816621546170426044088423240, 2.57356930524888605710413905810, 2.72124377519638254412480312830, 3.34932312375998979076903337646, 3.62707727887976064415127950913, 4.45743586498327452295321344360, 4.55845557930761765672675216511, 4.85158628959594320109486251729, 5.20984902525311716493396149347, 5.69442957324611895193295124170, 6.34434427454256381569107007670, 6.60504228364565900026710850533, 6.62336589275315757970408163728, 7.45285611450903001395411246290, 7.61911714559719413092592010498, 8.011448880203797533884299140563, 8.145520862771066887239374850421
