| L(s) = 1 | − 2·13-s − 19-s − 5·25-s − 7·31-s − 10·37-s − 5·43-s + 61-s + 16·67-s − 17·73-s + 4·79-s + 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.554·13-s − 0.229·19-s − 25-s − 1.25·31-s − 1.64·37-s − 0.762·43-s + 0.128·61-s + 1.95·67-s − 1.98·73-s + 0.450·79-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.319362002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319362002\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 17 T + p T^{2} \) | 1.73.r |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−15.66553243407727, −15.02970761703465, −14.47598075990227, −14.03648246143621, −13.41420768875524, −12.80586622757553, −12.40728460666403, −11.61934753489427, −11.39383557435307, −10.44691308920947, −10.17944538623827, −9.464055371270285, −8.888868885569261, −8.346699243517210, −7.595105091662441, −7.165699048080696, −6.491793974181406, −5.772426568368413, −5.202557527461710, −4.570429797897447, −3.736474285874611, −3.246184778600049, −2.195873901897803, −1.717436249156064, −0.4519342743888883,
0.4519342743888883, 1.717436249156064, 2.195873901897803, 3.246184778600049, 3.736474285874611, 4.570429797897447, 5.202557527461710, 5.772426568368413, 6.491793974181406, 7.165699048080696, 7.595105091662441, 8.346699243517210, 8.888868885569261, 9.464055371270285, 10.17944538623827, 10.44691308920947, 11.39383557435307, 11.61934753489427, 12.40728460666403, 12.80586622757553, 13.41420768875524, 14.03648246143621, 14.47598075990227, 15.02970761703465, 15.66553243407727
