| L(s) = 1 | + 2-s + 3-s + 6-s + 5·7-s − 8-s + 4·11-s − 2·13-s + 5·14-s − 16-s + 2·17-s − 19-s + 5·21-s + 4·22-s − 2·23-s − 24-s − 2·26-s − 27-s + 8·29-s + 4·33-s + 2·34-s + 3·37-s − 38-s − 2·39-s + 24·41-s + 5·42-s − 16·43-s − 2·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 1.20·11-s − 0.554·13-s + 1.33·14-s − 1/4·16-s + 0.485·17-s − 0.229·19-s + 1.09·21-s + 0.852·22-s − 0.417·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 1.48·29-s + 0.696·33-s + 0.342·34-s + 0.493·37-s − 0.162·38-s − 0.320·39-s + 3.74·41-s + 0.771·42-s − 2.43·43-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.205501876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.205501876\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−9.885402541530216065003883806669, −9.710295657921391609462350667901, −9.386260591263690445229289367104, −8.669389793481505132116177440196, −8.393791829652997053060695584901, −8.214843272729450627964479755401, −7.63393912654400044330095811871, −7.34575704552509271252165784771, −6.57803037360237864549288269393, −6.42786527467189148302351661791, −5.67426197382946463403343697812, −5.32332145085047594119610255988, −4.70736626634911097987615049385, −4.57209107021365080854470910218, −3.82154022124820655775315412367, −3.73877961714802317696799223125, −2.55557417139178764772792395553, −2.51914419268996084391199297535, −1.55159543623333042154548385138, −1.00012898623254346807919900663,
1.00012898623254346807919900663, 1.55159543623333042154548385138, 2.51914419268996084391199297535, 2.55557417139178764772792395553, 3.73877961714802317696799223125, 3.82154022124820655775315412367, 4.57209107021365080854470910218, 4.70736626634911097987615049385, 5.32332145085047594119610255988, 5.67426197382946463403343697812, 6.42786527467189148302351661791, 6.57803037360237864549288269393, 7.34575704552509271252165784771, 7.63393912654400044330095811871, 8.214843272729450627964479755401, 8.393791829652997053060695584901, 8.669389793481505132116177440196, 9.386260591263690445229289367104, 9.710295657921391609462350667901, 9.885402541530216065003883806669
