| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 2·11-s − 2·13-s + 15-s − 6·19-s + 2·21-s + 6·23-s + 25-s − 27-s − 6·29-s + 8·31-s + 2·33-s + 2·35-s − 6·37-s + 2·39-s − 6·41-s − 43-s − 45-s − 2·47-s − 3·49-s − 14·53-s + 2·55-s + 6·57-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.258·15-s − 1.37·19-s + 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.348·33-s + 0.338·35-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.152·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s − 1.92·53-s + 0.269·55-s + 0.794·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6208659201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6208659201\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.122968175095748289488355039023, −7.46676828890259221327353212057, −6.59360514399924771022956972178, −6.30536788734893702825727819203, −5.10116919600785733158992147776, −4.76498454682271098218000158559, −3.67622549313634534712182555086, −2.95165452398722071193364346552, −1.88076586693591689299995967288, −0.42194094716550452135701134875,
0.42194094716550452135701134875, 1.88076586693591689299995967288, 2.95165452398722071193364346552, 3.67622549313634534712182555086, 4.76498454682271098218000158559, 5.10116919600785733158992147776, 6.30536788734893702825727819203, 6.59360514399924771022956972178, 7.46676828890259221327353212057, 8.122968175095748289488355039023
