| L(s) = 1 | + 3·5-s + 2·7-s − 3·9-s − 2·13-s − 2·17-s + 4·19-s + 9·23-s + 4·25-s − 29-s + 4·31-s + 6·35-s + 8·37-s + 2·41-s − 43-s − 9·45-s + 10·47-s − 3·49-s + 9·53-s − 2·61-s − 6·63-s − 6·65-s + 14·67-s − 6·71-s − 4·73-s + 12·79-s + 9·81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.755·7-s − 9-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.87·23-s + 4/5·25-s − 0.185·29-s + 0.718·31-s + 1.01·35-s + 1.31·37-s + 0.312·41-s − 0.152·43-s − 1.34·45-s + 1.45·47-s − 3/7·49-s + 1.23·53-s − 0.256·61-s − 0.755·63-s − 0.744·65-s + 1.71·67-s − 0.712·71-s − 0.468·73-s + 1.35·79-s + 81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.701234533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.701234533\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3271 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.15586647030652, −12.65546356347429, −11.94210634954794, −11.64545570404621, −11.05043073382624, −10.82068218141900, −10.15086747515617, −9.710768956318922, −9.133667349120100, −8.980258108794891, −8.372696932318635, −7.714432802858751, −7.384032664102247, −6.630625701836211, −6.298050873607983, −5.618771290739568, −5.277718540165574, −4.932289593111704, −4.307602962630653, −3.503614949757852, −2.721113869562010, −2.556243347281389, −1.919083008649780, −1.113434998969616, −0.6659506252444294,
0.6659506252444294, 1.113434998969616, 1.919083008649780, 2.556243347281389, 2.721113869562010, 3.503614949757852, 4.307602962630653, 4.932289593111704, 5.277718540165574, 5.618771290739568, 6.298050873607983, 6.630625701836211, 7.384032664102247, 7.714432802858751, 8.372696932318635, 8.980258108794891, 9.133667349120100, 9.710768956318922, 10.15086747515617, 10.82068218141900, 11.05043073382624, 11.64545570404621, 11.94210634954794, 12.65546356347429, 13.15586647030652
