| L(s) = 1 | + 3-s − 7-s + 9-s − 4·13-s − 17-s − 4·19-s − 21-s + 27-s − 4·29-s − 10·31-s − 2·37-s − 4·39-s − 6·47-s + 49-s − 51-s + 2·53-s − 4·57-s − 4·59-s + 10·61-s − 63-s − 4·67-s + 12·71-s + 10·73-s − 2·79-s + 81-s − 6·83-s − 4·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.242·17-s − 0.917·19-s − 0.218·21-s + 0.192·27-s − 0.742·29-s − 1.79·31-s − 0.328·37-s − 0.640·39-s − 0.875·47-s + 1/7·49-s − 0.140·51-s + 0.274·53-s − 0.529·57-s − 0.520·59-s + 1.28·61-s − 0.125·63-s − 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.225·79-s + 1/9·81-s − 0.658·83-s − 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.322594797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322594797\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−14.94810193291775, −14.36397388901309, −14.06744840565300, −13.19014611514752, −12.86447392580539, −12.52616765687711, −11.79725936560352, −11.18658896459769, −10.65701102906197, −10.03561026409212, −9.559786785158054, −9.054302366600173, −8.559253103432750, −7.836695271519337, −7.377118828046565, −6.801824230318110, −6.276426444224846, −5.384787551454314, −4.991711080360787, −4.082897814535373, −3.692768762486912, −2.879971635888439, −2.208994700327025, −1.687088504972017, −0.3880762700396310,
0.3880762700396310, 1.687088504972017, 2.208994700327025, 2.879971635888439, 3.692768762486912, 4.082897814535373, 4.991711080360787, 5.384787551454314, 6.276426444224846, 6.801824230318110, 7.377118828046565, 7.836695271519337, 8.559253103432750, 9.054302366600173, 9.559786785158054, 10.03561026409212, 10.65701102906197, 11.18658896459769, 11.79725936560352, 12.52616765687711, 12.86447392580539, 13.19014611514752, 14.06744840565300, 14.36397388901309, 14.94810193291775
