| L(s) = 1 | − 2·4-s + 2·9-s − 4·13-s + 4·16-s − 6·17-s − 5·25-s − 4·36-s − 4·37-s − 6·41-s − 4·49-s + 8·52-s − 12·53-s + 20·61-s − 8·64-s + 12·68-s − 4·73-s − 5·81-s − 2·97-s + 10·100-s − 18·109-s + 6·113-s − 8·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s + 2/3·9-s − 1.10·13-s + 16-s − 1.45·17-s − 25-s − 2/3·36-s − 0.657·37-s − 0.937·41-s − 4/7·49-s + 1.10·52-s − 1.64·53-s + 2.56·61-s − 64-s + 1.45·68-s − 0.468·73-s − 5/9·81-s − 0.203·97-s + 100-s − 1.72·109-s + 0.564·113-s − 0.739·117-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.633731455286696412672852497783, −9.147133856531130991326216125738, −8.688905757931164651798661190122, −8.089547260783665697484307616611, −7.70530762752800765860050302538, −6.94464094352742810301507715699, −6.66178913992086289237323057036, −5.82912663564127308789458423232, −5.13850060431603881752548486460, −4.75654311786951999774401545118, −4.14464388430762590588758823830, −3.59605324322709519789951114412, −2.57808318130574018327169053328, −1.66683035103486635794989350528, 0,
1.66683035103486635794989350528, 2.57808318130574018327169053328, 3.59605324322709519789951114412, 4.14464388430762590588758823830, 4.75654311786951999774401545118, 5.13850060431603881752548486460, 5.82912663564127308789458423232, 6.66178913992086289237323057036, 6.94464094352742810301507715699, 7.70530762752800765860050302538, 8.089547260783665697484307616611, 8.688905757931164651798661190122, 9.147133856531130991326216125738, 9.633731455286696412672852497783
