| L(s) = 1 | + 2-s − 4-s − 5-s + 4·7-s − 3·8-s − 10-s + 2·11-s + 2·13-s + 4·14-s − 16-s + 17-s + 20-s + 2·22-s + 25-s + 2·26-s − 4·28-s + 6·29-s + 8·31-s + 5·32-s + 34-s − 4·35-s + 8·37-s + 3·40-s − 2·41-s − 2·43-s − 2·44-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s − 1.06·8-s − 0.316·10-s + 0.603·11-s + 0.554·13-s + 1.06·14-s − 1/4·16-s + 0.242·17-s + 0.223·20-s + 0.426·22-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.883·32-s + 0.171·34-s − 0.676·35-s + 1.31·37-s + 0.474·40-s − 0.312·41-s − 0.304·43-s − 0.301·44-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.061634832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.061634832\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−10.44787456578569469993098267623, −9.380761151050205211625571979315, −8.388802085021799096646662139328, −8.039484014745913836048377756002, −6.67603653027263764822986131544, −5.68500538082780401805673938737, −4.65554628665035948752787960158, −4.21354047156725909905435186301, −2.95336854432552375819757692984, −1.20847193299842148280252913365,
1.20847193299842148280252913365, 2.95336854432552375819757692984, 4.21354047156725909905435186301, 4.65554628665035948752787960158, 5.68500538082780401805673938737, 6.67603653027263764822986131544, 8.039484014745913836048377756002, 8.388802085021799096646662139328, 9.380761151050205211625571979315, 10.44787456578569469993098267623
