| L(s) = 1 | − 3-s + 7-s + 9-s + 2·13-s − 2·17-s − 4·19-s − 21-s − 27-s − 6·29-s + 6·37-s − 2·39-s − 6·41-s + 8·43-s − 8·47-s + 49-s + 2·51-s + 6·53-s + 4·57-s + 12·59-s − 10·61-s + 63-s + 16·67-s − 8·71-s + 6·73-s + 8·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.218·21-s − 0.192·27-s − 1.11·29-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s + 1.56·59-s − 1.28·61-s + 0.125·63-s + 1.95·67-s − 0.949·71-s + 0.702·73-s + 0.900·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−15.23756403268653, −14.80935724663983, −14.28290584431038, −13.54008390473481, −13.13589479799809, −12.68043376893474, −12.05031960207553, −11.45165083490013, −10.98283269894817, −10.74628197565161, −9.865566438543692, −9.506121213130487, −8.655094738609339, −8.340479656951136, −7.625497472994443, −6.955771122922671, −6.497250532884004, −5.796087262637367, −5.385082964962533, −4.552146219137258, −4.121456661971821, −3.433103714300592, −2.441717642276708, −1.827357316955692, −0.9656452119132939, 0,
0.9656452119132939, 1.827357316955692, 2.441717642276708, 3.433103714300592, 4.121456661971821, 4.552146219137258, 5.385082964962533, 5.796087262637367, 6.497250532884004, 6.955771122922671, 7.625497472994443, 8.340479656951136, 8.655094738609339, 9.506121213130487, 9.865566438543692, 10.74628197565161, 10.98283269894817, 11.45165083490013, 12.05031960207553, 12.68043376893474, 13.13589479799809, 13.54008390473481, 14.28290584431038, 14.80935724663983, 15.23756403268653
