| L(s) = 1 | − 2-s − 4-s − 4·7-s + 3·8-s + 9-s − 4·11-s + 4·14-s − 16-s − 18-s + 4·22-s − 8·23-s − 6·25-s + 4·28-s + 4·29-s − 5·32-s − 36-s − 12·37-s − 4·43-s + 4·44-s + 8·46-s + 9·49-s + 6·50-s + 4·53-s − 12·56-s − 4·58-s − 4·63-s + 7·64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 1.20·11-s + 1.06·14-s − 1/4·16-s − 0.235·18-s + 0.852·22-s − 1.66·23-s − 6/5·25-s + 0.755·28-s + 0.742·29-s − 0.883·32-s − 1/6·36-s − 1.97·37-s − 0.609·43-s + 0.603·44-s + 1.17·46-s + 9/7·49-s + 0.848·50-s + 0.549·53-s − 1.60·56-s − 0.525·58-s − 0.503·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 142 T^{2} + p^{2} T^{4} \) |
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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
−10.44454898913414486939724769503, −10.12029365328095173716368300113, −10.09990271855495778846661422563, −9.306204180494575020210566434453, −8.767444958989863355343652458642, −8.146135115788131129357222477812, −7.63787553506677429974008093288, −7.03947578447735256857195283239, −6.27855570452939790189820103933, −5.64719141855390848519374950583, −4.87603416006184254395760217623, −3.98440014921724214655734548220, −3.30940308562421900446742546394, −2.08203716194554985500996998790, 0,
2.08203716194554985500996998790, 3.30940308562421900446742546394, 3.98440014921724214655734548220, 4.87603416006184254395760217623, 5.64719141855390848519374950583, 6.27855570452939790189820103933, 7.03947578447735256857195283239, 7.63787553506677429974008093288, 8.146135115788131129357222477812, 8.767444958989863355343652458642, 9.306204180494575020210566434453, 10.09990271855495778846661422563, 10.12029365328095173716368300113, 10.44454898913414486939724769503
