| L(s) = 1 | − 3-s + 2·5-s − 7-s + 2·11-s − 6·13-s − 2·15-s − 8·17-s − 19-s + 21-s + 8·23-s + 5·25-s + 27-s + 8·29-s + 3·31-s − 2·33-s − 2·35-s + 37-s + 6·39-s + 12·41-s − 22·43-s + 6·47-s − 6·49-s + 8·51-s + 12·53-s + 4·55-s + 57-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 0.603·11-s − 1.66·13-s − 0.516·15-s − 1.94·17-s − 0.229·19-s + 0.218·21-s + 1.66·23-s + 25-s + 0.192·27-s + 1.48·29-s + 0.538·31-s − 0.348·33-s − 0.338·35-s + 0.164·37-s + 0.960·39-s + 1.87·41-s − 3.35·43-s + 0.875·47-s − 6/7·49-s + 1.12·51-s + 1.64·53-s + 0.539·55-s + 0.132·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.263153378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263153378\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 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 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−11.63790998096548664661348338424, −11.36765123625881381711486926441, −10.90929784649867002765494353685, −10.31442072471246883662636759286, −10.00197949853719039243880535707, −9.457551345235878957257558916503, −9.177044910692484362465839089257, −8.517582409680500408215238209730, −8.231584237362531551949448662763, −7.15074344755877876783206672592, −6.74208441855819844756001850826, −6.70317931633606283554179067551, −6.05827521831216853329198351574, −5.14749652134935296284852630223, −4.96898515500522395358742203573, −4.45672837578769976636787124889, −3.50617272240915416604153687559, −2.52854492416413074923365866843, −2.27091278224127219674876294817, −0.832988554727392064574717382835,
0.832988554727392064574717382835, 2.27091278224127219674876294817, 2.52854492416413074923365866843, 3.50617272240915416604153687559, 4.45672837578769976636787124889, 4.96898515500522395358742203573, 5.14749652134935296284852630223, 6.05827521831216853329198351574, 6.70317931633606283554179067551, 6.74208441855819844756001850826, 7.15074344755877876783206672592, 8.231584237362531551949448662763, 8.517582409680500408215238209730, 9.177044910692484362465839089257, 9.457551345235878957257558916503, 10.00197949853719039243880535707, 10.31442072471246883662636759286, 10.90929784649867002765494353685, 11.36765123625881381711486926441, 11.63790998096548664661348338424
