| L(s) = 1 | − 4·5-s − 4·13-s + 2·17-s + 11·25-s + 10·29-s − 2·37-s − 10·41-s − 7·49-s − 4·53-s − 12·61-s + 16·65-s + 16·73-s − 8·85-s + 16·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.10·13-s + 0.485·17-s + 11/5·25-s + 1.85·29-s − 0.328·37-s − 1.56·41-s − 49-s − 0.549·53-s − 1.53·61-s + 1.98·65-s + 1.87·73-s − 0.867·85-s + 1.69·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7617575312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7617575312\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−15.01942456830758, −14.53451503659997, −14.04877280805166, −13.36467552900695, −12.57480937606455, −12.24727906640895, −11.85063716680791, −11.46630883319120, −10.62759490683768, −10.36375213262470, −9.595826604153251, −8.974632179092060, −8.282319678384380, −7.914509545981503, −7.506553539498574, −6.740974170243103, −6.455708048232870, −5.274769197113624, −4.806413261328658, −4.394302923097009, −3.486510017791731, −3.189911359950677, −2.353866425433327, −1.255617976578253, −0.3524642558590171,
0.3524642558590171, 1.255617976578253, 2.353866425433327, 3.189911359950677, 3.486510017791731, 4.394302923097009, 4.806413261328658, 5.274769197113624, 6.455708048232870, 6.740974170243103, 7.506553539498574, 7.914509545981503, 8.282319678384380, 8.974632179092060, 9.595826604153251, 10.36375213262470, 10.62759490683768, 11.46630883319120, 11.85063716680791, 12.24727906640895, 12.57480937606455, 13.36467552900695, 14.04877280805166, 14.53451503659997, 15.01942456830758
