| L(s) = 1 | − 3-s − 4·7-s − 2·9-s + 6·11-s − 2·19-s + 4·21-s − 23-s + 5·27-s + 9·29-s − 5·31-s − 6·33-s + 2·37-s + 9·41-s + 4·43-s − 3·47-s + 9·49-s + 6·53-s + 2·57-s + 2·61-s + 8·63-s − 10·67-s + 69-s + 3·71-s − 7·73-s − 24·77-s − 10·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s − 2/3·9-s + 1.80·11-s − 0.458·19-s + 0.872·21-s − 0.208·23-s + 0.962·27-s + 1.67·29-s − 0.898·31-s − 1.04·33-s + 0.328·37-s + 1.40·41-s + 0.609·43-s − 0.437·47-s + 9/7·49-s + 0.824·53-s + 0.264·57-s + 0.256·61-s + 1.00·63-s − 1.22·67-s + 0.120·69-s + 0.356·71-s − 0.819·73-s − 2.73·77-s − 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.178813151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178813151\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.42966285143252, −11.94787638371293, −11.68205311461800, −11.16994585476810, −10.66384269388182, −10.19375056659715, −9.766933658942793, −9.256716724551748, −8.874661376440142, −8.621186673226580, −7.875752781377125, −7.188938510828758, −6.831522063884842, −6.361809626735184, −6.040837984030049, −5.767625506869369, −5.006122642891376, −4.317088501732948, −4.009406208914748, −3.463396169820001, −2.814517375017215, −2.531799426194106, −1.540758395057018, −0.9662441578693582, −0.3427434476584267,
0.3427434476584267, 0.9662441578693582, 1.540758395057018, 2.531799426194106, 2.814517375017215, 3.463396169820001, 4.009406208914748, 4.317088501732948, 5.006122642891376, 5.767625506869369, 6.040837984030049, 6.361809626735184, 6.831522063884842, 7.188938510828758, 7.875752781377125, 8.621186673226580, 8.874661376440142, 9.256716724551748, 9.766933658942793, 10.19375056659715, 10.66384269388182, 11.16994585476810, 11.68205311461800, 11.94787638371293, 12.42966285143252
