L(s) = 1 | − 2·7-s + 9-s + 2·11-s + 12·23-s + 2·25-s − 4·29-s − 8·37-s − 6·43-s − 3·49-s − 16·53-s − 2·63-s − 26·67-s − 4·71-s − 4·77-s + 81-s + 2·99-s − 14·107-s − 8·109-s − 8·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 1/3·9-s + 0.603·11-s + 2.50·23-s + 2/5·25-s − 0.742·29-s − 1.31·37-s − 0.914·43-s − 3/7·49-s − 2.19·53-s − 0.251·63-s − 3.17·67-s − 0.474·71-s − 0.455·77-s + 1/9·81-s + 0.201·99-s − 1.35·107-s − 0.766·109-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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−7.87702621073088423112387657093, −7.46168371035916923536010416084, −6.96360974643958694361366441090, −6.67486775112483790660623455339, −6.37528709623911141870620716717, −5.65688749634427664247643464794, −5.27454745022270864307069293970, −4.66393346445928918534684766138, −4.38397302800882760551193350306, −3.49473383869835185291054945217, −3.22295158536108118504664689946, −2.79454383998507020521271500231, −1.72007962845203144828908213712, −1.25817474054023838581546523424, 0,
1.25817474054023838581546523424, 1.72007962845203144828908213712, 2.79454383998507020521271500231, 3.22295158536108118504664689946, 3.49473383869835185291054945217, 4.38397302800882760551193350306, 4.66393346445928918534684766138, 5.27454745022270864307069293970, 5.65688749634427664247643464794, 6.37528709623911141870620716717, 6.67486775112483790660623455339, 6.96360974643958694361366441090, 7.46168371035916923536010416084, 7.87702621073088423112387657093