| L(s) = 1 | − 4·7-s − 9-s − 2·11-s + 4·23-s + 2·25-s − 10·29-s − 2·37-s + 18·43-s + 9·49-s − 2·53-s + 4·63-s − 10·67-s + 8·77-s + 20·79-s + 81-s + 2·99-s − 2·107-s + 30·109-s − 8·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1/3·9-s − 0.603·11-s + 0.834·23-s + 2/5·25-s − 1.85·29-s − 0.328·37-s + 2.74·43-s + 9/7·49-s − 0.274·53-s + 0.503·63-s − 1.22·67-s + 0.911·77-s + 2.25·79-s + 1/9·81-s + 0.201·99-s − 0.193·107-s + 2.87·109-s − 0.752·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.68445795534364367234465653663, −7.64239707402597781845155392289, −7.05481143963644260576475025515, −6.67824019095457632378705003299, −6.06827373428610273424602005934, −5.81609200307236074106263564170, −5.36336376153736675737515744860, −4.76942586208011197176770132300, −4.18902666237726665124292707777, −3.53431230838284831647074203740, −3.25496387908298583939377865267, −2.59689799218921971748024077375, −2.12018891361162115180966604330, −0.952931466335619982152361322813, 0,
0.952931466335619982152361322813, 2.12018891361162115180966604330, 2.59689799218921971748024077375, 3.25496387908298583939377865267, 3.53431230838284831647074203740, 4.18902666237726665124292707777, 4.76942586208011197176770132300, 5.36336376153736675737515744860, 5.81609200307236074106263564170, 6.06827373428610273424602005934, 6.67824019095457632378705003299, 7.05481143963644260576475025515, 7.64239707402597781845155392289, 7.68445795534364367234465653663
