| L(s) = 1 | + 2·13-s + 2·19-s + 2·31-s − 2·43-s − 2·49-s + 2·61-s + 2·67-s + 2·73-s − 2·79-s − 2·97-s − 2·103-s + 2·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·13-s + 2·19-s + 2·31-s − 2·43-s − 2·49-s + 2·61-s + 2·67-s + 2·73-s − 2·79-s − 2·97-s − 2·103-s + 2·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.910959325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910959325\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−8.490620138629912187602295426519, −8.476321267532536859773278556044, −8.153491483070236748947013663509, −8.027592019714757293011444457224, −7.27839556684461675343212061390, −6.94663146711433033383744186717, −6.59028095509068227918614996880, −6.38884549019917351630451837624, −5.83403724724397098352307289168, −5.52795646457976237600618673845, −4.98639311594012271599723071639, −4.92264760973941527807073366904, −4.14726335527498063426128062934, −3.80244944839643513579029235771, −3.39354447575815766800511358282, −3.07711542972640821109144637437, −2.58473831355688169607526086850, −1.84373994002479990322152495619, −1.26417555172452974123587518177, −0.934983145613013021130999173857,
0.934983145613013021130999173857, 1.26417555172452974123587518177, 1.84373994002479990322152495619, 2.58473831355688169607526086850, 3.07711542972640821109144637437, 3.39354447575815766800511358282, 3.80244944839643513579029235771, 4.14726335527498063426128062934, 4.92264760973941527807073366904, 4.98639311594012271599723071639, 5.52795646457976237600618673845, 5.83403724724397098352307289168, 6.38884549019917351630451837624, 6.59028095509068227918614996880, 6.94663146711433033383744186717, 7.27839556684461675343212061390, 8.027592019714757293011444457224, 8.153491483070236748947013663509, 8.476321267532536859773278556044, 8.490620138629912187602295426519
