| L(s) = 1 | − 9·13-s + 6·17-s − 8·19-s + 6·23-s + 5·25-s + 6·29-s + 2·31-s + 12·41-s − 9·43-s + 18·47-s + 11·49-s + 18·53-s − 12·59-s + 5·61-s − 13·67-s + 5·73-s + 79-s + 24·97-s − 2·103-s − 24·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.49·13-s + 1.45·17-s − 1.83·19-s + 1.25·23-s + 25-s + 1.11·29-s + 0.359·31-s + 1.87·41-s − 1.37·43-s + 2.62·47-s + 11/7·49-s + 2.47·53-s − 1.56·59-s + 0.640·61-s − 1.58·67-s + 0.585·73-s + 0.112·79-s + 2.43·97-s − 0.197·103-s − 2.29·109-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.403782818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.403782818\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 89 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T + 70 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 18 T + 155 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 18 T + 161 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−9.014214153243778854867179585060, −8.687060378136006594473080065015, −8.327453372297320095684435356319, −7.66350171950873883741614778833, −7.53848099665972617328565541188, −7.21966101702330435056828786085, −6.75174274607511829942234768190, −6.49430988263262441817710785932, −5.84211032669462278804462405690, −5.48002251931227763329425085367, −5.16546840831288235276245419748, −4.69493644329183855329332637383, −4.17151558810655660165013039338, −4.16682532921498144871198831288, −3.11494196354921670215982180470, −2.86831235986725589289400852387, −2.41461399522982433159967505647, −2.05495924288036352418961036127, −0.998996254448405603439831032123, −0.60542505835964058463003449133,
0.60542505835964058463003449133, 0.998996254448405603439831032123, 2.05495924288036352418961036127, 2.41461399522982433159967505647, 2.86831235986725589289400852387, 3.11494196354921670215982180470, 4.16682532921498144871198831288, 4.17151558810655660165013039338, 4.69493644329183855329332637383, 5.16546840831288235276245419748, 5.48002251931227763329425085367, 5.84211032669462278804462405690, 6.49430988263262441817710785932, 6.75174274607511829942234768190, 7.21966101702330435056828786085, 7.53848099665972617328565541188, 7.66350171950873883741614778833, 8.327453372297320095684435356319, 8.687060378136006594473080065015, 9.014214153243778854867179585060
