| L(s) = 1 | − 2·2-s − 13·4-s − 4·5-s − 14·7-s + 44·8-s + 8·10-s − 20·11-s + 32·13-s + 28·14-s + 101·16-s − 34·17-s + 36·19-s + 52·20-s + 40·22-s − 128·23-s − 238·25-s − 64·26-s + 182·28-s − 272·29-s + 100·31-s − 614·32-s + 68·34-s + 56·35-s − 584·37-s − 72·38-s − 176·40-s + 224·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·4-s − 0.357·5-s − 0.755·7-s + 1.94·8-s + 0.252·10-s − 0.548·11-s + 0.682·13-s + 0.534·14-s + 1.57·16-s − 0.485·17-s + 0.434·19-s + 0.581·20-s + 0.387·22-s − 1.16·23-s − 1.90·25-s − 0.482·26-s + 1.22·28-s − 1.74·29-s + 0.579·31-s − 3.39·32-s + 0.342·34-s + 0.270·35-s − 2.59·37-s − 0.307·38-s − 0.695·40-s + 0.853·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1147041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1147041 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4010478116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4010478116\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 20 T + 112 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 32 T + 3696 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 36 T - 3872 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 128 T + 21646 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 272 T + 62080 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 100 T + 41306 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 584 T + 173744 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 224 T - 19214 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 48 T + 132454 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 120 T + 150190 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 1116 T + 644848 T^{2} - 1116 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 60 T + 371758 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 568 T + 495198 T^{2} + 568 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1808 T + 1481734 T^{2} - 1808 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1488 T + 1270514 T^{2} + 1488 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 272 T + 67958 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1316 T + 1474672 T^{2} - 1316 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 436 T + 1270478 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1508 T + 2351462 T^{2} - 1508 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.570168580767799018017879918436, −9.437058013829631798957432869194, −8.859431818245228129982284090657, −8.556697314095017416219013132871, −8.075293358741733299592515573084, −7.942869001010087576596884271328, −7.21389920609860882217314272286, −7.17692779449754176710324683952, −6.05557802230514300221226974568, −6.03213537466727972413049593520, −5.29429978231595410430940068291, −5.08236445288907917132353789504, −4.36327748347560387020398853412, −3.83199663861468266330249045210, −3.71053398923614600150257615209, −3.21267146637638512850383801587, −2.01325821533073991710327715927, −1.75274311779951733878652244344, −0.65026664574733189124949333791, −0.30075092652064705173031267948,
0.30075092652064705173031267948, 0.65026664574733189124949333791, 1.75274311779951733878652244344, 2.01325821533073991710327715927, 3.21267146637638512850383801587, 3.71053398923614600150257615209, 3.83199663861468266330249045210, 4.36327748347560387020398853412, 5.08236445288907917132353789504, 5.29429978231595410430940068291, 6.03213537466727972413049593520, 6.05557802230514300221226974568, 7.17692779449754176710324683952, 7.21389920609860882217314272286, 7.942869001010087576596884271328, 8.075293358741733299592515573084, 8.556697314095017416219013132871, 8.859431818245228129982284090657, 9.437058013829631798957432869194, 9.570168580767799018017879918436
