| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s − 3·13-s + 16-s − 17-s − 18-s − 4·19-s + 22-s + 4·23-s + 24-s + 3·26-s − 27-s + 8·29-s + 5·31-s − 32-s + 33-s + 34-s + 36-s + 6·37-s + 4·38-s + 3·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.832·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s + 1.48·29-s + 0.898·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s + 0.986·37-s + 0.648·38-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−13.78472060037066, −13.09726147106387, −12.70334443435947, −12.23788690612854, −11.85896748770796, −11.16023925716495, −10.88372761862090, −10.38267668365176, −9.922179891642253, −9.454013147173896, −8.874132976176967, −8.449074562290564, −7.720222008285446, −7.504153178678259, −6.813108908919515, −6.170864391216403, −6.124836830855383, −5.124907493012431, −4.640993756075946, −4.340048244746603, −3.312520973866007, −2.686016408838151, −2.285554983455673, −1.374890591911795, −0.7506903207213823, 0,
0.7506903207213823, 1.374890591911795, 2.285554983455673, 2.686016408838151, 3.312520973866007, 4.340048244746603, 4.640993756075946, 5.124907493012431, 6.124836830855383, 6.170864391216403, 6.813108908919515, 7.504153178678259, 7.720222008285446, 8.449074562290564, 8.874132976176967, 9.454013147173896, 9.922179891642253, 10.38267668365176, 10.88372761862090, 11.16023925716495, 11.85896748770796, 12.23788690612854, 12.70334443435947, 13.09726147106387, 13.78472060037066
