| L(s) = 1 | + 4-s − 7-s − 2·13-s + 16-s − 13·19-s − 6·25-s − 28-s + 2·31-s + 18·37-s − 3·43-s − 6·49-s − 2·52-s − 9·61-s + 64-s − 12·73-s − 13·76-s − 4·79-s + 2·91-s − 7·97-s − 6·100-s − 14·103-s − 16·109-s − 112-s − 5·121-s + 2·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.377·7-s − 0.554·13-s + 1/4·16-s − 2.98·19-s − 6/5·25-s − 0.188·28-s + 0.359·31-s + 2.95·37-s − 0.457·43-s − 6/7·49-s − 0.277·52-s − 1.15·61-s + 1/8·64-s − 1.40·73-s − 1.49·76-s − 0.450·79-s + 0.209·91-s − 0.710·97-s − 3/5·100-s − 1.37·103-s − 1.53·109-s − 0.0944·112-s − 0.454·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 11 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.070423319071786879599192847940, −8.365991467442676807933913470480, −8.011081739965114407311446301572, −7.65539385532125537210998472476, −6.89183540690608546646615062738, −6.49265975997186221310376831873, −6.09303585608545649941337070442, −5.70942309555735719784658926784, −4.73640256764326308346091600610, −4.31157211219306238726407698773, −3.86577804668500197868712414454, −2.81405494667623761257166336037, −2.44173971628419086084364635003, −1.62023559930529901867186097008, 0,
1.62023559930529901867186097008, 2.44173971628419086084364635003, 2.81405494667623761257166336037, 3.86577804668500197868712414454, 4.31157211219306238726407698773, 4.73640256764326308346091600610, 5.70942309555735719784658926784, 6.09303585608545649941337070442, 6.49265975997186221310376831873, 6.89183540690608546646615062738, 7.65539385532125537210998472476, 8.011081739965114407311446301572, 8.365991467442676807933913470480, 9.070423319071786879599192847940
