| L(s) = 1 | − 6·5-s + 5·7-s + 6·11-s − 2·13-s + 3·17-s + 19-s − 6·23-s + 17·25-s − 6·29-s + 7·31-s − 30·35-s + 37-s + 6·41-s + 4·43-s − 9·47-s + 18·49-s + 3·53-s − 36·55-s + 9·59-s + 61-s + 12·65-s + 7·67-s + 73-s + 30·77-s + 13·79-s + 12·83-s − 18·85-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 1.88·7-s + 1.80·11-s − 0.554·13-s + 0.727·17-s + 0.229·19-s − 1.25·23-s + 17/5·25-s − 1.11·29-s + 1.25·31-s − 5.07·35-s + 0.164·37-s + 0.937·41-s + 0.609·43-s − 1.31·47-s + 18/7·49-s + 0.412·53-s − 4.85·55-s + 1.17·59-s + 0.128·61-s + 1.48·65-s + 0.855·67-s + 0.117·73-s + 3.41·77-s + 1.46·79-s + 1.31·83-s − 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113067823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113067823\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.026861867605260381224244182765, −8.686954904080750992429404424793, −8.208077386255098953944052598795, −8.046775282463125525462751855021, −7.70688963258776613259040676865, −7.57573961418871796428557587065, −6.98277802225329297155758503026, −6.77348592691730488382292906001, −6.01135986598653580852330347102, −5.72195888197127843599900900255, −4.94948962517530743792586606247, −4.73478362570569390012341664446, −4.26766695122424809613125007561, −4.02777428217235854159078374705, −3.55653432205470019691054163608, −3.38290175680366818695727836974, −2.30405514557169386913691066221, −1.90531808697282450301918967626, −1.02007408875464759529332644698, −0.64228351253592272391219834932,
0.64228351253592272391219834932, 1.02007408875464759529332644698, 1.90531808697282450301918967626, 2.30405514557169386913691066221, 3.38290175680366818695727836974, 3.55653432205470019691054163608, 4.02777428217235854159078374705, 4.26766695122424809613125007561, 4.73478362570569390012341664446, 4.94948962517530743792586606247, 5.72195888197127843599900900255, 6.01135986598653580852330347102, 6.77348592691730488382292906001, 6.98277802225329297155758503026, 7.57573961418871796428557587065, 7.70688963258776613259040676865, 8.046775282463125525462751855021, 8.208077386255098953944052598795, 8.686954904080750992429404424793, 9.026861867605260381224244182765
