| L(s) = 1 | + 3·7-s − 2·11-s − 13-s − 2·17-s − 5·19-s + 6·23-s + 10·29-s + 3·31-s − 2·37-s + 8·41-s + 43-s + 2·47-s + 2·49-s − 4·53-s + 10·59-s − 7·61-s − 3·67-s − 8·71-s − 14·73-s − 6·77-s − 6·83-s − 3·91-s + 17·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.603·11-s − 0.277·13-s − 0.485·17-s − 1.14·19-s + 1.25·23-s + 1.85·29-s + 0.538·31-s − 0.328·37-s + 1.24·41-s + 0.152·43-s + 0.291·47-s + 2/7·49-s − 0.549·53-s + 1.30·59-s − 0.896·61-s − 0.366·67-s − 0.949·71-s − 1.63·73-s − 0.683·77-s − 0.658·83-s − 0.314·91-s + 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.290338131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.290338131\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−15.89129277989968, −15.67162526634979, −14.84060891473335, −14.56569070891860, −13.97730307560571, −13.20647468735044, −12.87818321992124, −12.08267713362127, −11.62377727585948, −10.86893732894542, −10.59693050772993, −9.962056205374998, −8.999982194948345, −8.624324108512994, −8.037908280197421, −7.418575078694474, −6.756003046390375, −6.068787339110791, −5.288736144072255, −4.511462065169995, −4.428441197464709, −3.068357537539794, −2.497465835743630, −1.649315466739159, −0.6759803058628902,
0.6759803058628902, 1.649315466739159, 2.497465835743630, 3.068357537539794, 4.428441197464709, 4.511462065169995, 5.288736144072255, 6.068787339110791, 6.756003046390375, 7.418575078694474, 8.037908280197421, 8.624324108512994, 8.999982194948345, 9.962056205374998, 10.59693050772993, 10.86893732894542, 11.62377727585948, 12.08267713362127, 12.87818321992124, 13.20647468735044, 13.97730307560571, 14.56569070891860, 14.84060891473335, 15.67162526634979, 15.89129277989968
