| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 4·11-s + 6·13-s + 15-s + 6·17-s − 8·19-s − 21-s + 8·23-s + 25-s − 27-s + 2·29-s + 4·33-s − 35-s − 2·37-s − 6·39-s − 2·41-s + 8·43-s − 45-s + 12·47-s + 49-s − 6·51-s − 53-s + 4·55-s + 8·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s + 1.45·17-s − 1.83·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.169·35-s − 0.328·37-s − 0.960·39-s − 0.312·41-s + 1.21·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s − 0.840·51-s − 0.137·53-s + 0.539·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.296393861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.296393861\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 53 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.52389985214996, −12.29522968311740, −11.36097381257675, −11.17448868481325, −10.80597175330703, −10.39623186779578, −10.10151105535333, −9.222872952773275, −8.750383720508687, −8.507984168715236, −7.794129259324484, −7.662131252820431, −6.936117582241588, −6.507171087368701, −5.885615981177406, −5.592014020715680, −5.061955423683937, −4.501986947890689, −4.061779348134306, −3.471234146645903, −2.947468654905858, −2.351433230006210, −1.570043676324409, −0.9853235054194292, −0.4901397806190932,
0.4901397806190932, 0.9853235054194292, 1.570043676324409, 2.351433230006210, 2.947468654905858, 3.471234146645903, 4.061779348134306, 4.501986947890689, 5.061955423683937, 5.592014020715680, 5.885615981177406, 6.507171087368701, 6.936117582241588, 7.662131252820431, 7.794129259324484, 8.507984168715236, 8.750383720508687, 9.222872952773275, 10.10151105535333, 10.39623186779578, 10.80597175330703, 11.17448868481325, 11.36097381257675, 12.29522968311740, 12.52389985214996
