| L(s) = 1 | − 3-s − 2·7-s + 9-s − 13-s + 6·17-s − 19-s + 2·21-s − 6·23-s − 27-s + 7·29-s + 10·31-s − 7·37-s + 39-s + 3·41-s − 12·43-s − 5·47-s − 3·49-s − 6·51-s − 11·53-s + 57-s + 10·59-s + 2·61-s − 2·63-s + 11·67-s + 6·69-s − 5·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 1.45·17-s − 0.229·19-s + 0.436·21-s − 1.25·23-s − 0.192·27-s + 1.29·29-s + 1.79·31-s − 1.15·37-s + 0.160·39-s + 0.468·41-s − 1.82·43-s − 0.729·47-s − 3/7·49-s − 0.840·51-s − 1.51·53-s + 0.132·57-s + 1.30·59-s + 0.256·61-s − 0.251·63-s + 1.34·67-s + 0.722·69-s − 0.593·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.316291114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316291114\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.57486091427196, −12.99069052957051, −12.34503624863244, −12.12530437491930, −11.81219936792062, −11.14661024358340, −10.45987436720148, −10.07557070234195, −9.830449547372915, −9.368275544789751, −8.394143696495023, −8.174750027670921, −7.688799204351142, −6.845814287308109, −6.443636884081143, −6.250148974070407, −5.378442687209461, −5.038243246918106, −4.458742030597082, −3.665061722517205, −3.302332390079812, −2.623339034405535, −1.859708843426893, −1.109683029064635, −0.4003025555972680,
0.4003025555972680, 1.109683029064635, 1.859708843426893, 2.623339034405535, 3.302332390079812, 3.665061722517205, 4.458742030597082, 5.038243246918106, 5.378442687209461, 6.250148974070407, 6.443636884081143, 6.845814287308109, 7.688799204351142, 8.174750027670921, 8.394143696495023, 9.368275544789751, 9.830449547372915, 10.07557070234195, 10.45987436720148, 11.14661024358340, 11.81219936792062, 12.12530437491930, 12.34503624863244, 12.99069052957051, 13.57486091427196
