| L(s) = 1 | + 2·2-s − 4-s + 2·5-s + 2·7-s − 8·8-s + 4·10-s + 4·11-s + 4·13-s + 4·14-s − 7·16-s + 4·17-s − 10·19-s − 2·20-s + 8·22-s + 6·23-s + 3·25-s + 8·26-s − 2·28-s + 2·31-s + 14·32-s + 8·34-s + 4·35-s − 4·37-s − 20·38-s − 16·40-s + 20·41-s + 6·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 0.894·5-s + 0.755·7-s − 2.82·8-s + 1.26·10-s + 1.20·11-s + 1.10·13-s + 1.06·14-s − 7/4·16-s + 0.970·17-s − 2.29·19-s − 0.447·20-s + 1.70·22-s + 1.25·23-s + 3/5·25-s + 1.56·26-s − 0.377·28-s + 0.359·31-s + 2.47·32-s + 1.37·34-s + 0.676·35-s − 0.657·37-s − 3.24·38-s − 2.52·40-s + 3.12·41-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16040025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16040025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.256001184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.256001184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 26 T + 330 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 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
−8.606880773640420022587851611435, −8.412562461217224942454375282236, −7.84878681433484206733260444472, −7.81717801598152193977279362187, −6.65353216947917795506489890782, −6.64005090184271650166161614520, −6.23585113536776575337015092028, −6.11522748908649410404124329121, −5.47039277839543323618710483824, −5.26247903149933572576112398159, −4.80083994107509409611263718284, −4.56022684149831112228842100375, −3.95721314620239035764157317286, −3.94384514110674748840855310817, −3.34649522689398511533026626353, −2.95797725410996375178079025970, −2.34946494622149728176427039186, −1.79696239912496461110354482095, −1.12161819391266517728722113534, −0.66072745266850036704375554874,
0.66072745266850036704375554874, 1.12161819391266517728722113534, 1.79696239912496461110354482095, 2.34946494622149728176427039186, 2.95797725410996375178079025970, 3.34649522689398511533026626353, 3.94384514110674748840855310817, 3.95721314620239035764157317286, 4.56022684149831112228842100375, 4.80083994107509409611263718284, 5.26247903149933572576112398159, 5.47039277839543323618710483824, 6.11522748908649410404124329121, 6.23585113536776575337015092028, 6.64005090184271650166161614520, 6.65353216947917795506489890782, 7.81717801598152193977279362187, 7.84878681433484206733260444472, 8.412562461217224942454375282236, 8.606880773640420022587851611435
