| L(s) = 1 | − 3·11-s − 17-s + 25-s + 3·41-s − 15·43-s − 4·49-s + 27·59-s + 6·67-s − 12·73-s + 3·83-s − 3·89-s − 20·97-s + 15·107-s − 20·113-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 0.904·11-s − 0.242·17-s + 1/5·25-s + 0.468·41-s − 2.28·43-s − 4/7·49-s + 3.51·59-s + 0.733·67-s − 1.40·73-s + 0.329·83-s − 0.317·89-s − 2.03·97-s + 1.45·107-s − 1.88·113-s − 0.272·121-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 + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 69 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.85034675084394395092025141470, −7.12389070775462956756097283471, −6.84217080775839682778264004630, −6.60414263905518191794322330120, −5.73924444138764347643627650391, −5.58864494184883744569167788309, −5.08252327600198187727652677585, −4.58667777708792631066931083137, −4.13286994451750313209858823387, −3.46906886828170888033916453370, −3.07252528126448367814514603334, −2.39089552236733907061806607751, −1.93500589931458166489791424285, −1.02531001230794768063389608980, 0,
1.02531001230794768063389608980, 1.93500589931458166489791424285, 2.39089552236733907061806607751, 3.07252528126448367814514603334, 3.46906886828170888033916453370, 4.13286994451750313209858823387, 4.58667777708792631066931083137, 5.08252327600198187727652677585, 5.58864494184883744569167788309, 5.73924444138764347643627650391, 6.60414263905518191794322330120, 6.84217080775839682778264004630, 7.12389070775462956756097283471, 7.85034675084394395092025141470
