| L(s) = 1 | − 4·5-s + 8·11-s − 4·19-s + 11·25-s + 12·29-s − 12·31-s − 49-s − 32·55-s − 8·59-s − 4·61-s − 16·71-s + 32·79-s + 32·89-s + 16·95-s − 16·101-s − 36·109-s + 26·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + 149-s + 151-s + 48·155-s + 157-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2.41·11-s − 0.917·19-s + 11/5·25-s + 2.22·29-s − 2.15·31-s − 1/7·49-s − 4.31·55-s − 1.04·59-s − 0.512·61-s − 1.89·71-s + 3.60·79-s + 3.39·89-s + 1.64·95-s − 1.59·101-s − 3.44·109-s + 2.36·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 3.85·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.840646648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.840646648\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.383949834434229405597320807512, −7.913623775180740305041447614234, −7.81589703406006464654024524562, −7.44405477547172090249095034002, −6.76859367155054717686707958605, −6.64294914694382344016972255277, −6.54675306222340887426383423112, −6.08773216882187887150194051471, −5.36239487320947567970015956367, −5.10362541349360466206183278186, −4.48843887522150546993028800139, −4.22233960846044003554235007887, −4.03796273095759626209845775860, −3.63591265806906311126248800952, −3.14272197517303297972975012282, −2.91056700258392534705339202068, −1.98741969506551593509163250878, −1.62384918158256139771304552022, −0.953376686207962207846543021984, −0.45365333101092635130843334355,
0.45365333101092635130843334355, 0.953376686207962207846543021984, 1.62384918158256139771304552022, 1.98741969506551593509163250878, 2.91056700258392534705339202068, 3.14272197517303297972975012282, 3.63591265806906311126248800952, 4.03796273095759626209845775860, 4.22233960846044003554235007887, 4.48843887522150546993028800139, 5.10362541349360466206183278186, 5.36239487320947567970015956367, 6.08773216882187887150194051471, 6.54675306222340887426383423112, 6.64294914694382344016972255277, 6.76859367155054717686707958605, 7.44405477547172090249095034002, 7.81589703406006464654024524562, 7.913623775180740305041447614234, 8.383949834434229405597320807512
