| L(s) = 1 | + 6·7-s + 6·19-s + 2·25-s + 6·31-s − 4·37-s + 8·43-s + 14·49-s − 20·61-s + 2·67-s + 4·79-s + 8·103-s − 8·109-s − 10·121-s + 127-s + 131-s + 36·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 12·175-s + 179-s + ⋯ |
| L(s) = 1 | + 2.26·7-s + 1.37·19-s + 2/5·25-s + 1.07·31-s − 0.657·37-s + 1.21·43-s + 2·49-s − 2.56·61-s + 0.244·67-s + 0.450·79-s + 0.788·103-s − 0.766·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.0760·173-s + 0.907·175-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.120265992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.120265992\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.075257102118721441562989676744, −7.86756139395375305198988393851, −7.41778155909519335240701448999, −7.04107829495058172241014800754, −6.40290829742781820907724539550, −5.84122750036013329338006700633, −5.38931768486352998317478144494, −4.97504874090531212686413887179, −4.55334630045497318040636110189, −4.25197876941557033379623437614, −3.37613329269396856069104097184, −2.89486831361438852855982285933, −2.10529677189616315902088168255, −1.52669754955306856650657171662, −0.934741245515398847185651189012,
0.934741245515398847185651189012, 1.52669754955306856650657171662, 2.10529677189616315902088168255, 2.89486831361438852855982285933, 3.37613329269396856069104097184, 4.25197876941557033379623437614, 4.55334630045497318040636110189, 4.97504874090531212686413887179, 5.38931768486352998317478144494, 5.84122750036013329338006700633, 6.40290829742781820907724539550, 7.04107829495058172241014800754, 7.41778155909519335240701448999, 7.86756139395375305198988393851, 8.075257102118721441562989676744
