| L(s) = 1 | − 3-s − 2·5-s − 2·11-s − 2·13-s + 2·15-s − 19-s + 5·25-s + 27-s + 8·29-s − 9·31-s + 2·33-s − 3·37-s + 2·39-s + 20·41-s − 10·43-s + 6·47-s − 12·53-s + 4·55-s + 57-s + 12·59-s + 10·61-s + 4·65-s − 5·67-s + 12·71-s − 3·73-s − 5·75-s − 79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.603·11-s − 0.554·13-s + 0.516·15-s − 0.229·19-s + 25-s + 0.192·27-s + 1.48·29-s − 1.61·31-s + 0.348·33-s − 0.493·37-s + 0.320·39-s + 3.12·41-s − 1.52·43-s + 0.875·47-s − 1.64·53-s + 0.539·55-s + 0.132·57-s + 1.56·59-s + 1.28·61-s + 0.496·65-s − 0.610·67-s + 1.42·71-s − 0.351·73-s − 0.577·75-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138524994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138524994\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.095650843607342341286807058191, −8.684889161753761560473567019582, −8.480736461094823191957060942817, −7.909045691144720300120806875039, −7.63675229315057595033968778765, −7.32261834993700179439043149342, −6.85522947369371698647412011586, −6.57016608492475979686698495591, −5.86586756947129023559217404769, −5.84026189614859112568099672593, −4.96091405593089452204061687943, −4.95337912383628265493927035745, −4.52461183979568642344119666214, −3.84863333838841410091079739919, −3.55664990711288053508099302551, −2.99143157666048655603679958200, −2.40551943175937956168835354156, −2.01089182619269223536061129716, −0.958279965519298771505917870043, −0.47028297774634751305130093318,
0.47028297774634751305130093318, 0.958279965519298771505917870043, 2.01089182619269223536061129716, 2.40551943175937956168835354156, 2.99143157666048655603679958200, 3.55664990711288053508099302551, 3.84863333838841410091079739919, 4.52461183979568642344119666214, 4.95337912383628265493927035745, 4.96091405593089452204061687943, 5.84026189614859112568099672593, 5.86586756947129023559217404769, 6.57016608492475979686698495591, 6.85522947369371698647412011586, 7.32261834993700179439043149342, 7.63675229315057595033968778765, 7.909045691144720300120806875039, 8.480736461094823191957060942817, 8.684889161753761560473567019582, 9.095650843607342341286807058191
