| L(s) = 1 | − 3-s − 7-s + 9-s + 11-s − 4·13-s + 17-s + 21-s + 23-s − 27-s − 9·29-s − 2·31-s − 33-s + 3·37-s + 4·39-s + 4·41-s − 7·43-s + 4·47-s + 49-s − 51-s + 2·53-s − 12·59-s + 8·61-s − 63-s + 7·67-s − 69-s + 5·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.242·17-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.174·33-s + 0.493·37-s + 0.640·39-s + 0.624·41-s − 1.06·43-s + 0.583·47-s + 1/7·49-s − 0.140·51-s + 0.274·53-s − 1.56·59-s + 1.02·61-s − 0.125·63-s + 0.855·67-s − 0.120·69-s + 0.593·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.24664771464607, −14.71223224290228, −14.16675516080343, −13.60970252631797, −12.85622770050148, −12.62968927777788, −12.09070456951597, −11.40318639148790, −11.11661459173352, −10.40206010584529, −9.807748021734198, −9.452012435535964, −8.914080115976554, −8.050869368308718, −7.502630837324864, −7.043376970664986, −6.429323240082592, −5.813481085456443, −5.239034889239352, −4.726705427686998, −3.915790127507644, −3.413391517558518, −2.482055443630905, −1.862537261398996, −0.8525506725960056, 0,
0.8525506725960056, 1.862537261398996, 2.482055443630905, 3.413391517558518, 3.915790127507644, 4.726705427686998, 5.239034889239352, 5.813481085456443, 6.429323240082592, 7.043376970664986, 7.502630837324864, 8.050869368308718, 8.914080115976554, 9.452012435535964, 9.807748021734198, 10.40206010584529, 11.11661459173352, 11.40318639148790, 12.09070456951597, 12.62968927777788, 12.85622770050148, 13.60970252631797, 14.16675516080343, 14.71223224290228, 15.24664771464607
