| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s + 4·7-s + 3·9-s − 2·11-s + 2·12-s + 8·14-s + 16-s + 8·17-s + 6·18-s − 8·19-s + 8·21-s − 4·22-s + 8·23-s + 4·27-s + 4·28-s − 4·29-s − 2·32-s − 4·33-s + 16·34-s + 3·36-s − 12·37-s − 16·38-s + 4·41-s + 16·42-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 1.51·7-s + 9-s − 0.603·11-s + 0.577·12-s + 2.13·14-s + 1/4·16-s + 1.94·17-s + 1.41·18-s − 1.83·19-s + 1.74·21-s − 0.852·22-s + 1.66·23-s + 0.769·27-s + 0.755·28-s − 0.742·29-s − 0.353·32-s − 0.696·33-s + 2.74·34-s + 1/2·36-s − 1.97·37-s − 2.59·38-s + 0.624·41-s + 2.46·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.634494791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.634494791\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−10.82002632252892950333455169568, −10.11424095420408552862254576885, −9.308997396054922256355768701384, −9.286489844669918528238429666743, −8.480940432247750198962491246559, −8.453960338864933071540012523038, −7.79845806421108079384028164162, −7.37942904110561407914367273826, −7.32385730729931674703126760289, −6.43228708704950001734148242244, −5.80190799950083295532458278252, −5.35335886562641704644914626001, −4.95601741387159510689822654926, −4.67139490217686918531454533113, −3.95655058547620787825183083835, −3.78188653335801157699735763598, −3.08785192459668953939314963464, −2.49433343895989191011728466158, −1.86502780182333678808141367125, −1.16498376338234331325241217790,
1.16498376338234331325241217790, 1.86502780182333678808141367125, 2.49433343895989191011728466158, 3.08785192459668953939314963464, 3.78188653335801157699735763598, 3.95655058547620787825183083835, 4.67139490217686918531454533113, 4.95601741387159510689822654926, 5.35335886562641704644914626001, 5.80190799950083295532458278252, 6.43228708704950001734148242244, 7.32385730729931674703126760289, 7.37942904110561407914367273826, 7.79845806421108079384028164162, 8.453960338864933071540012523038, 8.480940432247750198962491246559, 9.286489844669918528238429666743, 9.308997396054922256355768701384, 10.11424095420408552862254576885, 10.82002632252892950333455169568
