| L(s) = 1 | − 2·5-s − 3·9-s − 6·13-s − 2·17-s − 25-s + 10·29-s + 2·37-s + 10·41-s + 6·45-s − 7·49-s − 14·53-s + 10·61-s + 12·65-s + 6·73-s + 9·81-s + 4·85-s − 10·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 18·117-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 1.66·13-s − 0.485·17-s − 1/5·25-s + 1.85·29-s + 0.328·37-s + 1.56·41-s + 0.894·45-s − 49-s − 1.92·53-s + 1.28·61-s + 1.48·65-s + 0.702·73-s + 81-s + 0.433·85-s − 1.05·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 31 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−15.44607872858117, −14.72149095760934, −14.33334928314550, −14.07494918929257, −13.13258322266304, −12.65910159079357, −12.01406977814206, −11.80776480537937, −11.12195926973933, −10.73265461533527, −9.865202476511329, −9.519361968419298, −8.815715444598358, −8.084090659455587, −7.924222595872994, −7.170846164722282, −6.574674592282918, −5.973341500947829, −5.147114745216748, −4.669619426095954, −4.125194444144228, −3.195719706008078, −2.719870593358961, −2.048939107892697, −0.7509516470224755, 0,
0.7509516470224755, 2.048939107892697, 2.719870593358961, 3.195719706008078, 4.125194444144228, 4.669619426095954, 5.147114745216748, 5.973341500947829, 6.574674592282918, 7.170846164722282, 7.924222595872994, 8.084090659455587, 8.815715444598358, 9.519361968419298, 9.865202476511329, 10.73265461533527, 11.12195926973933, 11.80776480537937, 12.01406977814206, 12.65910159079357, 13.13258322266304, 14.07494918929257, 14.33334928314550, 14.72149095760934, 15.44607872858117
