| L(s) = 1 | + 3·2-s + 3-s + 4·4-s + 3·6-s + 3·8-s + 3·11-s + 4·12-s + 7·13-s + 3·16-s + 9·22-s + 3·23-s + 3·24-s + 21·26-s − 27-s + 6·32-s + 3·33-s + 3·37-s + 7·39-s + 6·41-s + 10·43-s + 12·44-s + 9·46-s + 3·48-s − 7·49-s + 28·52-s − 3·54-s − 18·59-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 2·4-s + 1.22·6-s + 1.06·8-s + 0.904·11-s + 1.15·12-s + 1.94·13-s + 3/4·16-s + 1.91·22-s + 0.625·23-s + 0.612·24-s + 4.11·26-s − 0.192·27-s + 1.06·32-s + 0.522·33-s + 0.493·37-s + 1.12·39-s + 0.937·41-s + 1.52·43-s + 1.80·44-s + 1.32·46-s + 0.433·48-s − 49-s + 3.88·52-s − 0.408·54-s − 2.34·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.645121355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.645121355\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 91 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−10.33655290533496836420308194191, −9.843963231527390580171107357540, −9.105843186413900524784805238835, −9.069084932299466622369564730209, −8.639088792803456785164918736586, −8.046985918205892511596777274801, −7.55712781602263027829023812591, −7.26193385574861545151219010435, −6.31859453965618835958929932457, −6.30789815043422501137767938775, −5.88799104523779201314772109017, −5.46145510037921279529980404261, −4.65839595865025108296362679525, −4.52294054175957135050892081689, −3.88097013161846168618880854513, −3.70863208252683331751725555543, −3.01826139467454103950154576191, −2.79519202909103534337746912453, −1.67050637935550119428375772502, −1.15474611137418329177116412373,
1.15474611137418329177116412373, 1.67050637935550119428375772502, 2.79519202909103534337746912453, 3.01826139467454103950154576191, 3.70863208252683331751725555543, 3.88097013161846168618880854513, 4.52294054175957135050892081689, 4.65839595865025108296362679525, 5.46145510037921279529980404261, 5.88799104523779201314772109017, 6.30789815043422501137767938775, 6.31859453965618835958929932457, 7.26193385574861545151219010435, 7.55712781602263027829023812591, 8.046985918205892511596777274801, 8.639088792803456785164918736586, 9.069084932299466622369564730209, 9.105843186413900524784805238835, 9.843963231527390580171107357540, 10.33655290533496836420308194191
