| L(s) = 1 | − 2·5-s − 2·7-s − 9-s + 6·11-s + 6·13-s + 6·17-s + 4·19-s − 2·23-s − 2·25-s − 6·29-s + 4·35-s + 2·37-s + 2·41-s + 2·45-s − 6·49-s − 8·53-s − 12·55-s − 4·59-s + 4·61-s + 2·63-s − 12·65-s + 10·67-s − 20·71-s + 22·73-s − 12·77-s + 4·79-s − 8·81-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 1/3·9-s + 1.80·11-s + 1.66·13-s + 1.45·17-s + 0.917·19-s − 0.417·23-s − 2/5·25-s − 1.11·29-s + 0.676·35-s + 0.328·37-s + 0.312·41-s + 0.298·45-s − 6/7·49-s − 1.09·53-s − 1.61·55-s − 0.520·59-s + 0.512·61-s + 0.251·63-s − 1.48·65-s + 1.22·67-s − 2.37·71-s + 2.57·73-s − 1.36·77-s + 0.450·79-s − 8/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.464381712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464381712\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 154 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 237 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22 T + 247 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 22 T + 282 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 22 T + 270 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−11.44420022844019217459730267824, −11.43482725136845057376940017759, −10.93113266588058446465105268615, −10.14352922351397147710915189939, −9.778323752188661881278326986032, −9.199727331679352674644043276800, −9.082813866517354113111444850445, −8.332897996242843948381382374831, −7.74343865236245248842713373798, −7.69746742447534260949597711415, −6.63569228682367228543134706324, −6.54969789537431679782676605826, −5.81728743237325468371592233222, −5.53747273748160129156981598550, −4.51525184049940622871842639665, −3.87133324097187052973248182514, −3.44908455100663750040586254178, −3.26964018939303481314596199283, −1.77386441832074138732796153972, −0.908582126801032814230526486658,
0.908582126801032814230526486658, 1.77386441832074138732796153972, 3.26964018939303481314596199283, 3.44908455100663750040586254178, 3.87133324097187052973248182514, 4.51525184049940622871842639665, 5.53747273748160129156981598550, 5.81728743237325468371592233222, 6.54969789537431679782676605826, 6.63569228682367228543134706324, 7.69746742447534260949597711415, 7.74343865236245248842713373798, 8.332897996242843948381382374831, 9.082813866517354113111444850445, 9.199727331679352674644043276800, 9.778323752188661881278326986032, 10.14352922351397147710915189939, 10.93113266588058446465105268615, 11.43482725136845057376940017759, 11.44420022844019217459730267824
