| L(s) = 1 | + 3·5-s − 3·9-s − 3·11-s − 5·13-s − 6·17-s − 4·19-s + 8·23-s + 4·25-s + 8·29-s − 7·31-s + 37-s − 2·41-s − 6·43-s − 9·45-s + 6·47-s − 53-s − 9·55-s + 9·59-s + 6·61-s − 15·65-s − 11·67-s + 15·71-s − 2·73-s + 4·79-s + 9·81-s − 14·83-s − 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 9-s − 0.904·11-s − 1.38·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 4/5·25-s + 1.48·29-s − 1.25·31-s + 0.164·37-s − 0.312·41-s − 0.914·43-s − 1.34·45-s + 0.875·47-s − 0.137·53-s − 1.21·55-s + 1.17·59-s + 0.768·61-s − 1.86·65-s − 1.34·67-s + 1.78·71-s − 0.234·73-s + 0.450·79-s + 81-s − 1.53·83-s − 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9198695935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9198695935\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.41120357429384, −13.25438400172684, −12.78398556421055, −12.23386520519237, −11.64986287192037, −10.97777035850290, −10.71141620613769, −10.24608045283605, −9.626596082317962, −9.246290728627143, −8.654453958023975, −8.409537067236927, −7.618271885796331, −6.867927273108707, −6.717279479920992, −6.027218839241806, −5.373423484914698, −5.094040859558643, −4.657408549379299, −3.799888917554077, −2.839493854923900, −2.489981606625882, −2.252333709393300, −1.336674811328398, −0.2770131842925118,
0.2770131842925118, 1.336674811328398, 2.252333709393300, 2.489981606625882, 2.839493854923900, 3.799888917554077, 4.657408549379299, 5.094040859558643, 5.373423484914698, 6.027218839241806, 6.717279479920992, 6.867927273108707, 7.618271885796331, 8.409537067236927, 8.654453958023975, 9.246290728627143, 9.626596082317962, 10.24608045283605, 10.71141620613769, 10.97777035850290, 11.64986287192037, 12.23386520519237, 12.78398556421055, 13.25438400172684, 13.41120357429384
