| L(s) = 1 | + 2·3-s − 4·7-s + 9-s + 2·11-s − 2·13-s + 2·17-s − 2·19-s − 8·21-s + 4·23-s − 4·27-s − 6·29-s + 4·33-s − 10·37-s − 4·39-s − 6·41-s + 6·43-s − 8·47-s + 9·49-s + 4·51-s + 6·53-s − 4·57-s − 14·59-s + 2·61-s − 4·63-s + 10·67-s + 8·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.485·17-s − 0.458·19-s − 1.74·21-s + 0.834·23-s − 0.769·27-s − 1.11·29-s + 0.696·33-s − 1.64·37-s − 0.640·39-s − 0.937·41-s + 0.914·43-s − 1.16·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s − 0.529·57-s − 1.82·59-s + 0.256·61-s − 0.503·63-s + 1.22·67-s + 0.963·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.499715772662989764702573299468, −7.48649464154346783477619856196, −6.95023759239239106595773853946, −6.15302264435441050088112742528, −5.26498006511685590910060560318, −4.04799601522778258065319452547, −3.35101374838703231470604183982, −2.81830188953599087445367836313, −1.71401499382330216031350685870, 0,
1.71401499382330216031350685870, 2.81830188953599087445367836313, 3.35101374838703231470604183982, 4.04799601522778258065319452547, 5.26498006511685590910060560318, 6.15302264435441050088112742528, 6.95023759239239106595773853946, 7.48649464154346783477619856196, 8.499715772662989764702573299468
