L(s) = 1 | − 3-s − 2·4-s + 9-s + 2·12-s + 4·16-s − 2·19-s − 8·25-s − 27-s − 2·36-s − 8·43-s − 4·48-s + 6·49-s + 2·57-s − 8·64-s + 8·73-s + 8·75-s + 4·76-s + 81-s − 4·97-s + 16·100-s + 2·108-s − 22·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s + 0.577·12-s + 16-s − 0.458·19-s − 8/5·25-s − 0.192·27-s − 1/3·36-s − 1.21·43-s − 0.577·48-s + 6/7·49-s + 0.264·57-s − 64-s + 0.936·73-s + 0.923·75-s + 0.458·76-s + 1/9·81-s − 0.406·97-s + 8/5·100-s + 0.192·108-s − 2·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7463700685\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7463700685\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.265871408597079685582754808913, −8.051484312652988414365426757219, −7.60032108586395046526201131172, −6.94807080806951454321579456573, −6.59702482944648646906669127281, −5.94380444509347254456170620934, −5.63550741265522799013363256562, −5.18055404700896681250601151281, −4.62714274429773318593210680834, −4.15753377420270197311733196762, −3.74032582363132214339939969726, −3.13181751204204423587569297553, −2.22569202258500581525033334981, −1.51146695212089759558676726865, −0.47136295976735412272560845248,
0.47136295976735412272560845248, 1.51146695212089759558676726865, 2.22569202258500581525033334981, 3.13181751204204423587569297553, 3.74032582363132214339939969726, 4.15753377420270197311733196762, 4.62714274429773318593210680834, 5.18055404700896681250601151281, 5.63550741265522799013363256562, 5.94380444509347254456170620934, 6.59702482944648646906669127281, 6.94807080806951454321579456573, 7.60032108586395046526201131172, 8.051484312652988414365426757219, 8.265871408597079685582754808913