| L(s) = 1 | + 5-s + 3·11-s + 12·13-s + 5·17-s + 19-s − 7·23-s + 5·25-s − 4·29-s − 5·31-s − 3·37-s − 4·41-s − 8·43-s − 5·47-s − 53-s + 3·55-s − 15·59-s − 5·61-s + 12·65-s + 9·67-s + 7·73-s − 79-s + 24·83-s + 5·85-s − 7·89-s + 95-s + 4·97-s − 3·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.904·11-s + 3.32·13-s + 1.21·17-s + 0.229·19-s − 1.45·23-s + 25-s − 0.742·29-s − 0.898·31-s − 0.493·37-s − 0.624·41-s − 1.21·43-s − 0.729·47-s − 0.137·53-s + 0.404·55-s − 1.95·59-s − 0.640·61-s + 1.48·65-s + 1.09·67-s + 0.819·73-s − 0.112·79-s + 2.63·83-s + 0.542·85-s − 0.741·89-s + 0.102·95-s + 0.406·97-s − 0.298·101-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
| \(L(1)\) |
\(\approx\) |
\(4.227189267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.227189267\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 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
−8.630174664410185708837553370701, −8.565810706393070387180045289308, −8.038850944669312626590298515583, −7.80282716692242500201560986949, −7.23377403168723512277026530414, −6.80821318341198385062924458284, −6.30829239211167669393710486756, −6.25616953744981569696860124347, −5.76821767156460523402409760381, −5.62055055261425114368032867557, −4.92807142285422453828188220552, −4.61079654934476800978965472498, −3.86516695488715013432914872365, −3.60633756945297858719443904240, −3.43240027055457385093836139958, −3.05205070804364727168094531283, −1.85201547460898496370961848711, −1.81137935789367780370537998047, −1.27481569317139463838032250751, −0.66671496389935456606934437951,
0.66671496389935456606934437951, 1.27481569317139463838032250751, 1.81137935789367780370537998047, 1.85201547460898496370961848711, 3.05205070804364727168094531283, 3.43240027055457385093836139958, 3.60633756945297858719443904240, 3.86516695488715013432914872365, 4.61079654934476800978965472498, 4.92807142285422453828188220552, 5.62055055261425114368032867557, 5.76821767156460523402409760381, 6.25616953744981569696860124347, 6.30829239211167669393710486756, 6.80821318341198385062924458284, 7.23377403168723512277026530414, 7.80282716692242500201560986949, 8.038850944669312626590298515583, 8.565810706393070387180045289308, 8.630174664410185708837553370701
