| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s + 2·13-s + 14-s − 16-s − 2·17-s + 4·19-s + 6·23-s − 2·26-s + 28-s + 4·29-s + 4·31-s − 5·32-s + 2·34-s + 6·37-s − 4·38-s − 4·43-s − 6·46-s + 6·47-s + 49-s − 2·52-s − 6·53-s − 3·56-s − 4·58-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 1.25·23-s − 0.392·26-s + 0.188·28-s + 0.742·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.648·38-s − 0.609·43-s − 0.884·46-s + 0.875·47-s + 1/7·49-s − 0.277·52-s − 0.824·53-s − 0.400·56-s − 0.525·58-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944021549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944021549\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.01810883133189, −12.88799434250805, −12.10499317988242, −11.63711819431634, −11.07876874128634, −10.75211415476260, −10.06384717430123, −9.852715464394335, −9.274186853864456, −8.761787017961484, −8.606538764926438, −7.811182852470353, −7.544615586536540, −6.899109329293960, −6.403834562889021, −5.905270617165776, −5.084491813244631, −4.868322768275223, −4.213462909797700, −3.603536092667392, −3.083916821918361, −2.430078356702189, −1.645200607031049, −0.8643204887879986, −0.6275204113055514,
0.6275204113055514, 0.8643204887879986, 1.645200607031049, 2.430078356702189, 3.083916821918361, 3.603536092667392, 4.213462909797700, 4.868322768275223, 5.084491813244631, 5.905270617165776, 6.403834562889021, 6.899109329293960, 7.544615586536540, 7.811182852470353, 8.606538764926438, 8.761787017961484, 9.274186853864456, 9.852715464394335, 10.06384717430123, 10.75211415476260, 11.07876874128634, 11.63711819431634, 12.10499317988242, 12.88799434250805, 13.01810883133189
