| L(s) = 1 | + 4-s − 4·7-s − 3·9-s − 3·16-s + 12·17-s − 4·28-s − 3·36-s + 4·37-s − 12·41-s + 16·43-s − 24·47-s + 9·49-s + 24·59-s + 12·63-s − 7·64-s − 16·67-s + 12·68-s + 16·79-s + 9·81-s − 12·89-s + 12·101-s − 4·109-s + 12·112-s − 48·119-s + 10·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.51·7-s − 9-s − 3/4·16-s + 2.91·17-s − 0.755·28-s − 1/2·36-s + 0.657·37-s − 1.87·41-s + 2.43·43-s − 3.50·47-s + 9/7·49-s + 3.12·59-s + 1.51·63-s − 7/8·64-s − 1.95·67-s + 1.45·68-s + 1.80·79-s + 81-s − 1.27·89-s + 1.19·101-s − 0.383·109-s + 1.13·112-s − 4.40·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.361520203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.361520203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−11.45237090828949478474383705406, −10.47212270757445484131359350354, −10.04587377959947312920249244919, −9.977409902473546394359670122745, −9.225366481388965813566345209463, −9.133659500153687179359894363497, −8.162909626428848534701027055385, −8.111675385615524935608401307889, −7.47384913185477231515421335849, −6.76658042870903684333855826712, −6.69271271891388967550066955704, −5.87330229878917318829235879409, −5.71045363267218455269869724935, −5.17036163867387101142343237463, −4.36762313245250054001214211565, −3.46471743218789052078383852691, −3.25680015186414588701100659352, −2.79266163078902945878075225686, −1.88052529093017967831170943536, −0.68009710406838669263553681590,
0.68009710406838669263553681590, 1.88052529093017967831170943536, 2.79266163078902945878075225686, 3.25680015186414588701100659352, 3.46471743218789052078383852691, 4.36762313245250054001214211565, 5.17036163867387101142343237463, 5.71045363267218455269869724935, 5.87330229878917318829235879409, 6.69271271891388967550066955704, 6.76658042870903684333855826712, 7.47384913185477231515421335849, 8.111675385615524935608401307889, 8.162909626428848534701027055385, 9.133659500153687179359894363497, 9.225366481388965813566345209463, 9.977409902473546394359670122745, 10.04587377959947312920249244919, 10.47212270757445484131359350354, 11.45237090828949478474383705406
