| L(s) = 1 | + 3-s + 4·7-s + 9-s + 3·11-s + 13-s − 7·17-s + 4·19-s + 4·21-s − 7·23-s − 5·25-s + 27-s − 4·29-s − 3·31-s + 3·33-s − 4·37-s + 39-s + 9·41-s − 12·47-s + 9·49-s − 7·51-s + 5·53-s + 4·57-s + 8·61-s + 4·63-s − 7·67-s − 7·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 1.69·17-s + 0.917·19-s + 0.872·21-s − 1.45·23-s − 25-s + 0.192·27-s − 0.742·29-s − 0.538·31-s + 0.522·33-s − 0.657·37-s + 0.160·39-s + 1.40·41-s − 1.75·47-s + 9/7·49-s − 0.980·51-s + 0.686·53-s + 0.529·57-s + 1.02·61-s + 0.503·63-s − 0.855·67-s − 0.842·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.694006649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.694006649\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−12.48258691067427, −11.96415252908143, −11.56881345894978, −11.23477644440691, −10.89899793722856, −10.27134528241697, −9.588846500092847, −9.433426799369873, −8.797207461146337, −8.391103183695034, −8.073719031826315, −7.486303165322366, −7.172722217676229, −6.534018919890060, −5.998992347069947, −5.525516386091695, −4.900731377777548, −4.432734720431727, −3.900571580085337, −3.712360975123367, −2.811154146331646, −2.031533012373299, −1.894757512115627, −1.340390018898540, −0.4569352455013308,
0.4569352455013308, 1.340390018898540, 1.894757512115627, 2.031533012373299, 2.811154146331646, 3.712360975123367, 3.900571580085337, 4.432734720431727, 4.900731377777548, 5.525516386091695, 5.998992347069947, 6.534018919890060, 7.172722217676229, 7.486303165322366, 8.073719031826315, 8.391103183695034, 8.797207461146337, 9.433426799369873, 9.588846500092847, 10.27134528241697, 10.89899793722856, 11.23477644440691, 11.56881345894978, 11.96415252908143, 12.48258691067427
