| L(s) = 1 | + 3-s − 3·7-s + 9-s − 2·11-s − 2·13-s + 4·17-s − 19-s − 3·21-s + 2·23-s + 27-s + 3·29-s + 4·31-s − 2·33-s + 6·37-s − 2·39-s − 11·41-s + 4·43-s + 2·49-s + 4·51-s − 3·53-s − 57-s − 3·59-s − 5·61-s − 3·63-s + 8·67-s + 2·69-s − 13·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.970·17-s − 0.229·19-s − 0.654·21-s + 0.417·23-s + 0.192·27-s + 0.557·29-s + 0.718·31-s − 0.348·33-s + 0.986·37-s − 0.320·39-s − 1.71·41-s + 0.609·43-s + 2/7·49-s + 0.560·51-s − 0.412·53-s − 0.132·57-s − 0.390·59-s − 0.640·61-s − 0.377·63-s + 0.977·67-s + 0.240·69-s − 1.54·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.81963730970936059578028045733, −7.08872951260381855959204260692, −6.44766491865973585229368328360, −5.64958533068185298562522714525, −4.82280856510239801671923202234, −3.97013236365828763110886675257, −2.98181547134595409374287927353, −2.72634849940879601105414336208, −1.36183508250428162060624918445, 0,
1.36183508250428162060624918445, 2.72634849940879601105414336208, 2.98181547134595409374287927353, 3.97013236365828763110886675257, 4.82280856510239801671923202234, 5.64958533068185298562522714525, 6.44766491865973585229368328360, 7.08872951260381855959204260692, 7.81963730970936059578028045733
