L(s) = 1 | + 2·3-s − 4-s − 7-s + 9-s − 2·12-s − 2·13-s + 16-s − 2·21-s + 2·25-s − 4·27-s + 28-s − 8·31-s − 36-s + 16·37-s − 4·39-s − 14·43-s + 2·48-s + 49-s + 2·52-s + 12·61-s − 63-s − 64-s − 73-s + 4·75-s − 4·79-s − 11·81-s + 2·84-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1/2·4-s − 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.554·13-s + 1/4·16-s − 0.436·21-s + 2/5·25-s − 0.769·27-s + 0.188·28-s − 1.43·31-s − 1/6·36-s + 2.63·37-s − 0.640·39-s − 2.13·43-s + 0.288·48-s + 1/7·49-s + 0.277·52-s + 1.53·61-s − 0.125·63-s − 1/8·64-s − 0.117·73-s + 0.461·75-s − 0.450·79-s − 1.22·81-s + 0.218·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.101451523332059389441414220934, −7.53012283520830470672949513330, −7.24678266930791089497711344659, −6.78209539771573183653717009592, −6.04087840134255496261575183416, −5.85922803970098985033481789005, −4.98869808527250273453434285248, −4.87339579371396446561727960108, −3.99236403007372923060728658470, −3.75080368839514761585140542836, −3.14472260752924686000824118892, −2.60651825696654980537491098183, −2.13082877707974009998466704081, −1.21766576142469790564065907187, 0,
1.21766576142469790564065907187, 2.13082877707974009998466704081, 2.60651825696654980537491098183, 3.14472260752924686000824118892, 3.75080368839514761585140542836, 3.99236403007372923060728658470, 4.87339579371396446561727960108, 4.98869808527250273453434285248, 5.85922803970098985033481789005, 6.04087840134255496261575183416, 6.78209539771573183653717009592, 7.24678266930791089497711344659, 7.53012283520830470672949513330, 8.101451523332059389441414220934