| L(s) = 1 | + 88·11-s − 88·19-s + 396·29-s − 320·31-s + 396·41-s + 110·49-s − 1.33e3·59-s + 1.10e3·61-s − 1.45e3·71-s + 1.31e3·79-s + 1.42e3·89-s − 3.13e3·101-s + 3.98e3·109-s + 3.14e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.91e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 1.06·19-s + 2.53·29-s − 1.85·31-s + 1.50·41-s + 0.320·49-s − 2.94·59-s + 2.30·61-s − 2.43·71-s + 1.86·79-s + 1.70·89-s − 3.08·101-s + 3.50·109-s + 2.36·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.77·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.622890080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.622890080\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 110 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3910 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7326 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21198 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 198 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 160 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 75062 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 198 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156310 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 71138 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 239190 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 668 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 550 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 566182 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 728 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 754318 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 656 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1087878 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 714 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1596862 T^{2} + p^{6} T^{4} \) |
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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
−9.048473101418463889075286577985, −8.886397274718240434098090120120, −8.287837908537914846137721612328, −8.142298061756749312682345713711, −7.33168026022743649229506987330, −7.18693297031578068531369002674, −6.61459630006631799007986852518, −6.40800205177388487579591912584, −5.97598244050244866584389143352, −5.67624485213076056463636725734, −4.84084873907269866422430375346, −4.55607896592534203082858991708, −4.11766051955143675120745147198, −3.81428952123884947843552335356, −3.19401233373564570325358541259, −2.77431112236465035751427785598, −1.90477565237156927213514079134, −1.71372946539629422514491691474, −0.909564284672483593749011255220, −0.55729554055613311136005269880,
0.55729554055613311136005269880, 0.909564284672483593749011255220, 1.71372946539629422514491691474, 1.90477565237156927213514079134, 2.77431112236465035751427785598, 3.19401233373564570325358541259, 3.81428952123884947843552335356, 4.11766051955143675120745147198, 4.55607896592534203082858991708, 4.84084873907269866422430375346, 5.67624485213076056463636725734, 5.97598244050244866584389143352, 6.40800205177388487579591912584, 6.61459630006631799007986852518, 7.18693297031578068531369002674, 7.33168026022743649229506987330, 8.142298061756749312682345713711, 8.287837908537914846137721612328, 8.886397274718240434098090120120, 9.048473101418463889075286577985
