| L(s) = 1 | + 4·5-s + 7-s − 4·13-s − 17-s + 8·19-s − 3·23-s + 11·25-s + 29-s + 4·35-s + 7·37-s + 9·41-s − 43-s + 5·47-s − 6·49-s − 4·53-s + 5·59-s + 8·61-s − 16·65-s − 8·67-s + 12·71-s − 4·73-s + 11·79-s + 12·83-s − 4·85-s − 4·89-s − 4·91-s + 32·95-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s − 1.10·13-s − 0.242·17-s + 1.83·19-s − 0.625·23-s + 11/5·25-s + 0.185·29-s + 0.676·35-s + 1.15·37-s + 1.40·41-s − 0.152·43-s + 0.729·47-s − 6/7·49-s − 0.549·53-s + 0.650·59-s + 1.02·61-s − 1.98·65-s − 0.977·67-s + 1.42·71-s − 0.468·73-s + 1.23·79-s + 1.31·83-s − 0.433·85-s − 0.423·89-s − 0.419·91-s + 3.28·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.287011281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.287011281\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.39972674182022, −14.01662244588875, −13.43531973340302, −13.17103036165649, −12.31398294214544, −12.13735968409598, −11.26410042624144, −10.89782060110857, −10.12915397369118, −9.759341124417230, −9.425772229292032, −9.015070904342325, −8.041718304517183, −7.721737073177928, −6.950195170918173, −6.500709983101524, −5.742957529519501, −5.451718870111855, −4.891966137157103, −4.287647125238055, −3.301733785190437, −2.566190883531400, −2.231001013306752, −1.418789801015932, −0.7467167753539376,
0.7467167753539376, 1.418789801015932, 2.231001013306752, 2.566190883531400, 3.301733785190437, 4.287647125238055, 4.891966137157103, 5.451718870111855, 5.742957529519501, 6.500709983101524, 6.950195170918173, 7.721737073177928, 8.041718304517183, 9.015070904342325, 9.425772229292032, 9.759341124417230, 10.12915397369118, 10.89782060110857, 11.26410042624144, 12.13735968409598, 12.31398294214544, 13.17103036165649, 13.43531973340302, 14.01662244588875, 14.39972674182022
