| L(s) = 1 | + 2·3-s + 5-s + 3·7-s + 9-s − 3·11-s − 4·13-s + 2·15-s + 5·17-s + 6·21-s − 4·25-s − 4·27-s + 2·29-s + 8·31-s − 6·33-s + 3·35-s − 10·37-s − 8·39-s − 6·41-s − 7·43-s + 45-s + 9·47-s + 2·49-s + 10·51-s − 8·53-s − 3·55-s − 14·59-s + 5·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.516·15-s + 1.21·17-s + 1.30·21-s − 4/5·25-s − 0.769·27-s + 0.371·29-s + 1.43·31-s − 1.04·33-s + 0.507·35-s − 1.64·37-s − 1.28·39-s − 0.937·41-s − 1.06·43-s + 0.149·45-s + 1.31·47-s + 2/7·49-s + 1.40·51-s − 1.09·53-s − 0.404·55-s − 1.82·59-s + 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 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
−15.50056375545883, −15.12300693020017, −14.57442240291786, −14.10840755145883, −13.75715562389216, −13.32527273643694, −12.42149825262587, −12.02131368367538, −11.50640246931258, −10.61705502831035, −10.08622765391477, −9.820496649820573, −8.971069706865481, −8.467287163406758, −7.848125940335852, −7.700370978050024, −6.923722403675670, −5.977853854264242, −5.285455866103631, −4.907476858799743, −4.131943291905665, −3.140155096846263, −2.795420780060002, −1.959213407238686, −1.454818415578364, 0,
1.454818415578364, 1.959213407238686, 2.795420780060002, 3.140155096846263, 4.131943291905665, 4.907476858799743, 5.285455866103631, 5.977853854264242, 6.923722403675670, 7.700370978050024, 7.848125940335852, 8.467287163406758, 8.971069706865481, 9.820496649820573, 10.08622765391477, 10.61705502831035, 11.50640246931258, 12.02131368367538, 12.42149825262587, 13.32527273643694, 13.75715562389216, 14.10840755145883, 14.57442240291786, 15.12300693020017, 15.50056375545883
