| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 2·11-s − 7·13-s + 2·14-s + 16-s − 4·17-s + 19-s − 2·22-s + 2·23-s + 7·26-s − 2·28-s − 4·29-s − 4·31-s − 32-s + 4·34-s − 4·37-s − 38-s − 5·41-s − 7·43-s + 2·44-s − 2·46-s − 47-s − 3·49-s − 7·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.603·11-s − 1.94·13-s + 0.534·14-s + 1/4·16-s − 0.970·17-s + 0.229·19-s − 0.426·22-s + 0.417·23-s + 1.37·26-s − 0.377·28-s − 0.742·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.657·37-s − 0.162·38-s − 0.780·41-s − 1.06·43-s + 0.301·44-s − 0.294·46-s − 0.145·47-s − 3/7·49-s − 0.970·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 9 T + p T^{2} \) | 1.61.j |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−15.45578556418624, −14.83327532644943, −14.74943147270051, −13.83986850116218, −13.35025810224081, −12.66985723389767, −12.28946874198227, −11.76843042985892, −11.17739561091386, −10.63086446661452, −9.935002758181464, −9.621973752410842, −9.106052391130303, −8.680715683621508, −7.774315300082138, −7.364163004905401, −6.796330472257348, −6.408925171108669, −5.603738273245040, −4.883909598994602, −4.375585905342237, −3.321875588092629, −2.982674878719580, −2.044094507650111, −1.515490889024060, 0, 0,
1.515490889024060, 2.044094507650111, 2.982674878719580, 3.321875588092629, 4.375585905342237, 4.883909598994602, 5.603738273245040, 6.408925171108669, 6.796330472257348, 7.364163004905401, 7.774315300082138, 8.680715683621508, 9.106052391130303, 9.621973752410842, 9.935002758181464, 10.63086446661452, 11.17739561091386, 11.76843042985892, 12.28946874198227, 12.66985723389767, 13.35025810224081, 13.83986850116218, 14.74943147270051, 14.83327532644943, 15.45578556418624
