| L(s) = 1 | − 3·5-s + 2·11-s + 5·13-s − 4·17-s + 6·19-s + 23-s + 4·25-s − 29-s − 4·31-s + 37-s + 41-s − 43-s + 9·47-s + 6·53-s − 6·55-s − 8·59-s + 12·61-s − 15·65-s + 16·67-s + 10·73-s − 6·79-s − 12·83-s + 12·85-s − 4·89-s − 18·95-s + 3·97-s + 101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.603·11-s + 1.38·13-s − 0.970·17-s + 1.37·19-s + 0.208·23-s + 4/5·25-s − 0.185·29-s − 0.718·31-s + 0.164·37-s + 0.156·41-s − 0.152·43-s + 1.31·47-s + 0.824·53-s − 0.809·55-s − 1.04·59-s + 1.53·61-s − 1.86·65-s + 1.95·67-s + 1.17·73-s − 0.675·79-s − 1.31·83-s + 1.30·85-s − 0.423·89-s − 1.84·95-s + 0.304·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−13.57761076456110, −13.00639515339227, −12.46864681129718, −12.02894372602768, −11.49622988681797, −11.18272391325445, −10.93773221841781, −10.25734336581430, −9.482939673185785, −9.221751627378400, −8.544170509708215, −8.303846155923947, −7.709294169398561, −7.112199032214008, −6.866978919097034, −6.196952116239516, −5.568915711039244, −5.116937755511786, −4.310660294648778, −3.832668338657279, −3.696575154167999, −2.930279289194157, −2.239070270313649, −1.307359819092299, −0.8736235140987493, 0,
0.8736235140987493, 1.307359819092299, 2.239070270313649, 2.930279289194157, 3.696575154167999, 3.832668338657279, 4.310660294648778, 5.116937755511786, 5.568915711039244, 6.196952116239516, 6.866978919097034, 7.112199032214008, 7.709294169398561, 8.303846155923947, 8.544170509708215, 9.221751627378400, 9.482939673185785, 10.25734336581430, 10.93773221841781, 11.18272391325445, 11.49622988681797, 12.02894372602768, 12.46864681129718, 13.00639515339227, 13.57761076456110
