| L(s) = 1 | − 6·7-s + 4·11-s − 13-s + 4·17-s − 8·19-s + 4·23-s + 5·25-s − 14·31-s − 2·37-s + 4·41-s + 43-s + 2·47-s + 13·49-s + 12·53-s + 2·59-s + 61-s + 13·67-s − 10·71-s + 3·73-s − 24·77-s − 13·79-s − 16·83-s + 18·89-s + 6·91-s − 2·97-s − 2·101-s − 30·103-s + ⋯ |
| L(s) = 1 | − 2.26·7-s + 1.20·11-s − 0.277·13-s + 0.970·17-s − 1.83·19-s + 0.834·23-s + 25-s − 2.51·31-s − 0.328·37-s + 0.624·41-s + 0.152·43-s + 0.291·47-s + 13/7·49-s + 1.64·53-s + 0.260·59-s + 0.128·61-s + 1.58·67-s − 1.18·71-s + 0.351·73-s − 2.73·77-s − 1.46·79-s − 1.75·83-s + 1.90·89-s + 0.628·91-s − 0.203·97-s − 0.199·101-s − 2.95·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9540549762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9540549762\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.232474864439347505165282036136, −8.773434273092903486015006743940, −8.358115171615758353057889440961, −7.921041143249434289003514573985, −7.23111397888516264958985378214, −6.95010924339115518519024910872, −6.81485429721530988393428180408, −6.51994874673751362855577306813, −5.88582151073364376686962739379, −5.67269891625774088879714764840, −5.33287101841860637752901723327, −4.57236452608051412545273378512, −4.10188887272556364424689172778, −3.79207092640190020104386394738, −3.41787794535285027578752582287, −2.94626411658241489702309749377, −2.52194225676281867381747078295, −1.84913486102734847549861541025, −1.14158653700534608631061445766, −0.34361747882934839941131248295,
0.34361747882934839941131248295, 1.14158653700534608631061445766, 1.84913486102734847549861541025, 2.52194225676281867381747078295, 2.94626411658241489702309749377, 3.41787794535285027578752582287, 3.79207092640190020104386394738, 4.10188887272556364424689172778, 4.57236452608051412545273378512, 5.33287101841860637752901723327, 5.67269891625774088879714764840, 5.88582151073364376686962739379, 6.51994874673751362855577306813, 6.81485429721530988393428180408, 6.95010924339115518519024910872, 7.23111397888516264958985378214, 7.921041143249434289003514573985, 8.358115171615758353057889440961, 8.773434273092903486015006743940, 9.232474864439347505165282036136
