| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 3·11-s − 2·13-s − 2·14-s + 16-s − 3·17-s + 3·22-s − 6·23-s − 5·25-s + 2·26-s + 2·28-s − 6·29-s + 4·31-s − 32-s + 3·34-s + 4·37-s − 9·41-s − 43-s − 3·44-s + 6·46-s − 6·47-s − 3·49-s + 5·50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 0.904·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.639·22-s − 1.25·23-s − 25-s + 0.392·26-s + 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.514·34-s + 0.657·37-s − 1.40·41-s − 0.152·43-s − 0.452·44-s + 0.884·46-s − 0.875·47-s − 3/7·49-s + 0.707·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58482 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58482 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3709588291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3709588291\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−14.29041634596018, −13.94471310555416, −13.26980465933413, −12.86541756378935, −12.12624245245753, −11.74547348625731, −11.17062504455917, −10.83647746794617, −10.16417056120835, −9.638473625877378, −9.408884545090327, −8.405280378525666, −8.090830016122359, −7.822684696900045, −7.144295010183943, −6.453523041740764, −6.009465225277305, −5.130551577704727, −4.901267852021809, −4.026889673489285, −3.400291661962049, −2.464587562904059, −2.064557181499252, −1.435718367765437, −0.2239896087096854,
0.2239896087096854, 1.435718367765437, 2.064557181499252, 2.464587562904059, 3.400291661962049, 4.026889673489285, 4.901267852021809, 5.130551577704727, 6.009465225277305, 6.453523041740764, 7.144295010183943, 7.822684696900045, 8.090830016122359, 8.405280378525666, 9.408884545090327, 9.638473625877378, 10.16417056120835, 10.83647746794617, 11.17062504455917, 11.74547348625731, 12.12624245245753, 12.86541756378935, 13.26980465933413, 13.94471310555416, 14.29041634596018
