| L(s) = 1 | − 3-s + 7-s − 2·9-s + 4·11-s + 13-s − 17-s + 6·19-s − 21-s + 5·27-s + 7·31-s − 4·33-s + 4·37-s − 39-s − 2·41-s + 4·43-s − 6·47-s − 6·49-s + 51-s − 11·53-s − 6·57-s − 8·59-s + 10·61-s − 2·63-s + 8·67-s − 7·71-s − 4·73-s + 4·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.20·11-s + 0.277·13-s − 0.242·17-s + 1.37·19-s − 0.218·21-s + 0.962·27-s + 1.25·31-s − 0.696·33-s + 0.657·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.875·47-s − 6/7·49-s + 0.140·51-s − 1.51·53-s − 0.794·57-s − 1.04·59-s + 1.28·61-s − 0.251·63-s + 0.977·67-s − 0.830·71-s − 0.468·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.796755543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.796755543\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 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.ah |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.046711301639399596916554123287, −7.19145298376916179047365397233, −6.38671630639600937963870420667, −6.01545261579017102801571670117, −5.08840369959606090935834967277, −4.55346366719129408738575943973, −3.55761606791559378659420145671, −2.83832178803906560836186038687, −1.60763108315141353265305040310, −0.75051932536388600511539141931,
0.75051932536388600511539141931, 1.60763108315141353265305040310, 2.83832178803906560836186038687, 3.55761606791559378659420145671, 4.55346366719129408738575943973, 5.08840369959606090935834967277, 6.01545261579017102801571670117, 6.38671630639600937963870420667, 7.19145298376916179047365397233, 8.046711301639399596916554123287
