L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 2·12-s − 13-s − 3·23-s + 4·27-s + 3·29-s + 3·36-s − 2·39-s + 2·43-s − 49-s − 52-s + 2·61-s − 64-s − 6·69-s − 79-s + 5·81-s + 6·87-s − 3·92-s − 103-s + 4·108-s + 3·116-s − 3·117-s + 121-s + 127-s + 4·129-s + ⋯ |
L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 2·12-s − 13-s − 3·23-s + 4·27-s + 3·29-s + 3·36-s − 2·39-s + 2·43-s − 49-s − 52-s + 2·61-s − 64-s − 6·69-s − 79-s + 5·81-s + 6·87-s − 3·92-s − 103-s + 4·108-s + 3·116-s − 3·117-s + 121-s + 127-s + 4·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.809749958\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.809749958\) |
\(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.011803129509337660087032524685, −8.652413244334396453544107483757, −8.312374516755770733425426926118, −8.012099382562616019296108178265, −7.73503466082006036443695530623, −7.35271946575011852512331160784, −6.96838665869194703548953374107, −6.64540413559197328850293174871, −6.20777470872312083621191869065, −5.94119926150063309653908041131, −5.10014227738254434226044130620, −4.68804668415531114189549644890, −4.32192066677180653193422690428, −3.87156994043140486267481632491, −3.56335907794005351837963543553, −2.74032081412975021635178061459, −2.49831245392009793428463772958, −2.43476575950368154038989713057, −1.74083266992747888707149059598, −1.16407494743949634103978667390,
1.16407494743949634103978667390, 1.74083266992747888707149059598, 2.43476575950368154038989713057, 2.49831245392009793428463772958, 2.74032081412975021635178061459, 3.56335907794005351837963543553, 3.87156994043140486267481632491, 4.32192066677180653193422690428, 4.68804668415531114189549644890, 5.10014227738254434226044130620, 5.94119926150063309653908041131, 6.20777470872312083621191869065, 6.64540413559197328850293174871, 6.96838665869194703548953374107, 7.35271946575011852512331160784, 7.73503466082006036443695530623, 8.012099382562616019296108178265, 8.312374516755770733425426926118, 8.652413244334396453544107483757, 9.011803129509337660087032524685