| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 9-s + 4·13-s + 5·16-s − 2·18-s + 8·26-s − 8·29-s + 6·32-s − 3·36-s + 8·37-s − 14·49-s + 12·52-s − 16·58-s − 20·61-s + 7·64-s + 20·67-s − 4·72-s + 12·73-s + 16·74-s + 16·79-s + 81-s + 28·97-s − 28·98-s + 16·101-s + 16·104-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1/3·9-s + 1.10·13-s + 5/4·16-s − 0.471·18-s + 1.56·26-s − 1.48·29-s + 1.06·32-s − 1/2·36-s + 1.31·37-s − 2·49-s + 1.66·52-s − 2.10·58-s − 2.56·61-s + 7/8·64-s + 2.44·67-s − 0.471·72-s + 1.40·73-s + 1.85·74-s + 1.80·79-s + 1/9·81-s + 2.84·97-s − 2.82·98-s + 1.59·101-s + 1.56·104-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.362753682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.362753682\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.434667563293574858454104998240, −8.886523840401281914411206428382, −8.696552147254585597012943373384, −7.84954046133758381871875453885, −7.78903191326942455805947105579, −7.57890789636835864850367380507, −6.73097409306985938747973742310, −6.32348519188408125550787316452, −6.32264056210951013726041810822, −5.82462603755263349736329311202, −5.14582123041252896236355918626, −5.08255065135544625293200762185, −4.53127134664982239360982595946, −3.79297305775890104753365929140, −3.76623472439731720679553981675, −3.21226331271031054578421884867, −2.69536341333025241394149460749, −2.04031592071084337002022513943, −1.61752912988400627581928835745, −0.71478162549208498852725680244,
0.71478162549208498852725680244, 1.61752912988400627581928835745, 2.04031592071084337002022513943, 2.69536341333025241394149460749, 3.21226331271031054578421884867, 3.76623472439731720679553981675, 3.79297305775890104753365929140, 4.53127134664982239360982595946, 5.08255065135544625293200762185, 5.14582123041252896236355918626, 5.82462603755263349736329311202, 6.32264056210951013726041810822, 6.32348519188408125550787316452, 6.73097409306985938747973742310, 7.57890789636835864850367380507, 7.78903191326942455805947105579, 7.84954046133758381871875453885, 8.696552147254585597012943373384, 8.886523840401281914411206428382, 9.434667563293574858454104998240
