| L(s) = 1 | + 4-s − 6·9-s − 8·11-s − 2·29-s + 16·31-s − 6·36-s + 18·41-s − 8·44-s + 2·49-s − 8·59-s − 14·61-s − 64-s + 16·71-s + 16·79-s + 9·81-s − 12·89-s + 48·99-s − 14·101-s + 8·109-s − 2·116-s + 38·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2·9-s − 2.41·11-s − 0.371·29-s + 2.87·31-s − 36-s + 2.81·41-s − 1.20·44-s + 2/7·49-s − 1.04·59-s − 1.79·61-s − 1/8·64-s + 1.89·71-s + 1.80·79-s + 81-s − 1.27·89-s + 4.82·99-s − 1.39·101-s + 0.766·109-s − 0.185·116-s + 3.45·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.587485794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587485794\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64440090888471205407183644984, −7.47057953950097738401686145619, −7.17761209985030149240914205021, −6.96572641400873621483563268841, −6.51742077064137406483133723657, −6.43385728576875241311207717413, −6.12461220682501026577217382826, −5.79446905463061659546830680489, −5.60034404424060112891284323826, −5.56532112037077169937601262187, −5.51483680838494261696577765261, −4.71423993774995886462195233697, −4.60682463175509487754318673477, −4.51518566006225694065999702056, −4.46428528797228893896671213373, −3.59453650167956852239802233846, −3.31839395665777794749042032370, −3.27107762032616699035558959007, −2.73724946181445990725714237399, −2.68037382483886837285955979260, −2.33821431991323016906059493486, −2.23814912793082852776572003796, −1.57556347262722484911703511943, −0.78846958689503377595506759044, −0.46029968845804465031127256987,
0.46029968845804465031127256987, 0.78846958689503377595506759044, 1.57556347262722484911703511943, 2.23814912793082852776572003796, 2.33821431991323016906059493486, 2.68037382483886837285955979260, 2.73724946181445990725714237399, 3.27107762032616699035558959007, 3.31839395665777794749042032370, 3.59453650167956852239802233846, 4.46428528797228893896671213373, 4.51518566006225694065999702056, 4.60682463175509487754318673477, 4.71423993774995886462195233697, 5.51483680838494261696577765261, 5.56532112037077169937601262187, 5.60034404424060112891284323826, 5.79446905463061659546830680489, 6.12461220682501026577217382826, 6.43385728576875241311207717413, 6.51742077064137406483133723657, 6.96572641400873621483563268841, 7.17761209985030149240914205021, 7.47057953950097738401686145619, 7.64440090888471205407183644984
