L(s) = 1 | + 2·3-s − 2·4-s + 9-s + 4·11-s − 4·12-s + 4·16-s + 2·19-s + 6·25-s − 4·27-s + 8·33-s − 2·36-s + 8·41-s − 14·43-s − 8·44-s + 8·48-s + 2·49-s + 4·57-s + 10·59-s − 8·64-s + 2·67-s + 20·73-s + 12·75-s − 4·76-s − 11·81-s − 20·89-s + 12·97-s + 4·99-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 4-s + 1/3·9-s + 1.20·11-s − 1.15·12-s + 16-s + 0.458·19-s + 6/5·25-s − 0.769·27-s + 1.39·33-s − 1/3·36-s + 1.24·41-s − 2.13·43-s − 1.20·44-s + 1.15·48-s + 2/7·49-s + 0.529·57-s + 1.30·59-s − 64-s + 0.244·67-s + 2.34·73-s + 1.38·75-s − 0.458·76-s − 1.22·81-s − 2.11·89-s + 1.21·97-s + 0.402·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.775513951\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.775513951\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 113 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.763115112618116926593583340462, −9.253244850303103632918268422399, −8.934378157250735909274858290334, −8.405625494704305756939963067305, −8.164755510833238577774451528619, −7.42587913865188688795426729358, −6.89971322470879843858479366659, −6.28522008363674035684659848112, −5.54148455756633746835211789311, −4.94951204053151806021768680507, −4.29236972126143639377325873081, −3.62849897297158305526116002325, −3.25972353252661926041149408314, −2.28649166833768613469376324136, −1.13863093393384090818968127410,
1.13863093393384090818968127410, 2.28649166833768613469376324136, 3.25972353252661926041149408314, 3.62849897297158305526116002325, 4.29236972126143639377325873081, 4.94951204053151806021768680507, 5.54148455756633746835211789311, 6.28522008363674035684659848112, 6.89971322470879843858479366659, 7.42587913865188688795426729358, 8.164755510833238577774451528619, 8.405625494704305756939963067305, 8.934378157250735909274858290334, 9.253244850303103632918268422399, 9.763115112618116926593583340462