| L(s) = 1 | + 7-s − 5·11-s − 6·13-s − 7·17-s − 6·19-s + 2·23-s − 8·31-s − 8·37-s − 8·41-s + 43-s − 8·47-s − 6·49-s + 3·53-s − 4·59-s + 11·61-s − 67-s + 15·71-s + 4·73-s − 5·77-s − 16·83-s − 2·89-s − 6·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.50·11-s − 1.66·13-s − 1.69·17-s − 1.37·19-s + 0.417·23-s − 1.43·31-s − 1.31·37-s − 1.24·41-s + 0.152·43-s − 1.16·47-s − 6/7·49-s + 0.412·53-s − 0.520·59-s + 1.40·61-s − 0.122·67-s + 1.78·71-s + 0.468·73-s − 0.569·77-s − 1.75·83-s − 0.211·89-s − 0.628·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−15.81433854854494, −15.30766016142869, −14.97441512676472, −14.44660891790859, −13.75699388056656, −13.10615598761012, −12.78828714933758, −12.34340946234677, −11.48388678364705, −10.99523781031088, −10.58275652063573, −9.970367195936702, −9.413843375747077, −8.584971549212875, −8.316841588370769, −7.573916920453696, −6.909194026578516, −6.637020180258046, −5.497169882961472, −5.066621139496848, −4.648451954005399, −3.845502885274559, −2.887670105649176, −2.230800230487833, −1.833851997379005, 0, 0,
1.833851997379005, 2.230800230487833, 2.887670105649176, 3.845502885274559, 4.648451954005399, 5.066621139496848, 5.497169882961472, 6.637020180258046, 6.909194026578516, 7.573916920453696, 8.316841588370769, 8.584971549212875, 9.413843375747077, 9.970367195936702, 10.58275652063573, 10.99523781031088, 11.48388678364705, 12.34340946234677, 12.78828714933758, 13.10615598761012, 13.75699388056656, 14.44660891790859, 14.97441512676472, 15.30766016142869, 15.81433854854494
