| L(s) = 1 | + 2·5-s − 4·11-s + 4·13-s − 6·17-s − 8·19-s + 5·25-s − 12·29-s − 8·31-s + 2·37-s − 4·41-s − 8·43-s − 8·47-s + 6·53-s − 8·55-s + 6·61-s + 8·65-s + 4·67-s + 16·71-s − 10·73-s − 16·79-s − 16·83-s − 12·85-s − 6·89-s − 16·95-s − 12·97-s + 2·101-s + 16·103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s + 1.10·13-s − 1.45·17-s − 1.83·19-s + 25-s − 2.22·29-s − 1.43·31-s + 0.328·37-s − 0.624·41-s − 1.21·43-s − 1.16·47-s + 0.824·53-s − 1.07·55-s + 0.768·61-s + 0.992·65-s + 0.488·67-s + 1.89·71-s − 1.17·73-s − 1.80·79-s − 1.75·83-s − 1.30·85-s − 0.635·89-s − 1.64·95-s − 1.21·97-s + 0.199·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2510755741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2510755741\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.788359837733482487556373952670, −8.407017069926959385513442878512, −8.242792853811672089924177324000, −7.62135792691190498769032676722, −7.18966060393141447416276885459, −6.83686683140558046296564218602, −6.57266779395104694305806105163, −6.10721642595436179116271076953, −5.75636346312353936597940121682, −5.36124022439734986559795617703, −5.15880243154139159184087563759, −4.38682891005626220835059926523, −4.31809853736027382662252033508, −3.51889863051200712724345189545, −3.44222563005466661107446109977, −2.41984590654836277065955255392, −2.40472872855139262739252710613, −1.77459012431860518593929449632, −1.42328680425892955404019516207, −0.13794029871789830177843569088,
0.13794029871789830177843569088, 1.42328680425892955404019516207, 1.77459012431860518593929449632, 2.40472872855139262739252710613, 2.41984590654836277065955255392, 3.44222563005466661107446109977, 3.51889863051200712724345189545, 4.31809853736027382662252033508, 4.38682891005626220835059926523, 5.15880243154139159184087563759, 5.36124022439734986559795617703, 5.75636346312353936597940121682, 6.10721642595436179116271076953, 6.57266779395104694305806105163, 6.83686683140558046296564218602, 7.18966060393141447416276885459, 7.62135792691190498769032676722, 8.242792853811672089924177324000, 8.407017069926959385513442878512, 8.788359837733482487556373952670
