| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 4·7-s − 8-s + 9-s − 4·11-s + 12-s − 4·14-s + 16-s − 4·17-s − 18-s − 7·19-s + 4·21-s + 4·22-s + 4·23-s − 24-s + 27-s + 4·28-s + 5·29-s − 4·31-s − 32-s − 4·33-s + 4·34-s + 36-s − 9·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.60·19-s + 0.872·21-s + 0.852·22-s + 0.834·23-s − 0.204·24-s + 0.192·27-s + 0.755·28-s + 0.928·29-s − 0.718·31-s − 0.176·32-s − 0.696·33-s + 0.685·34-s + 1/6·36-s − 1.47·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801915408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801915408\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−15.37739097491459, −14.87102947656113, −14.46142420013576, −13.80304474930754, −13.14437252445432, −12.78546281546932, −12.04368050986940, −11.29640308044221, −10.97420076428458, −10.43522093993666, −10.02455296995505, −9.066961987714363, −8.509513174277362, −8.391575339476529, −7.793215207825983, −7.000835618991253, −6.718398983434509, −5.632536752233184, −5.020340718520190, −4.527486613890115, −3.719227032964508, −2.746138847282986, −2.138737124194310, −1.703363722993133, −0.5548480554930588,
0.5548480554930588, 1.703363722993133, 2.138737124194310, 2.746138847282986, 3.719227032964508, 4.527486613890115, 5.020340718520190, 5.632536752233184, 6.718398983434509, 7.000835618991253, 7.793215207825983, 8.391575339476529, 8.509513174277362, 9.066961987714363, 10.02455296995505, 10.43522093993666, 10.97420076428458, 11.29640308044221, 12.04368050986940, 12.78546281546932, 13.14437252445432, 13.80304474930754, 14.46142420013576, 14.87102947656113, 15.37739097491459
