| L(s) = 1 | − 5-s − 5·7-s − 5·11-s + 17-s + 3·19-s − 3·23-s + 25-s + 29-s + 5·35-s − 7·37-s − 5·41-s + 5·43-s + 12·47-s + 18·49-s − 2·53-s + 5·55-s − 11·59-s − 13·61-s − 3·67-s + 13·71-s + 2·73-s + 25·77-s − 4·79-s + 12·83-s − 85-s + 7·89-s − 3·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.88·7-s − 1.50·11-s + 0.242·17-s + 0.688·19-s − 0.625·23-s + 1/5·25-s + 0.185·29-s + 0.845·35-s − 1.15·37-s − 0.780·41-s + 0.762·43-s + 1.75·47-s + 18/7·49-s − 0.274·53-s + 0.674·55-s − 1.43·59-s − 1.66·61-s − 0.366·67-s + 1.54·71-s + 0.234·73-s + 2.84·77-s − 0.450·79-s + 1.31·83-s − 0.108·85-s + 0.741·89-s − 0.307·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−15.48240765259828, −15.18986167297631, −14.11643477129164, −13.70066238136261, −13.33359344278700, −12.61491599534378, −12.29158602451798, −11.93616390804848, −10.87808277594902, −10.51440237014006, −10.11095559108905, −9.385216319009909, −9.080860951477840, −8.198682714790579, −7.674655856465492, −7.181568038290319, −6.559304827189664, −5.891784196585304, −5.446171369405011, −4.672649462335357, −3.861805274645379, −3.209861953088509, −2.881631211835971, −2.025826280236902, −0.7042592455587949, 0,
0.7042592455587949, 2.025826280236902, 2.881631211835971, 3.209861953088509, 3.861805274645379, 4.672649462335357, 5.446171369405011, 5.891784196585304, 6.559304827189664, 7.181568038290319, 7.674655856465492, 8.198682714790579, 9.080860951477840, 9.385216319009909, 10.11095559108905, 10.51440237014006, 10.87808277594902, 11.93616390804848, 12.29158602451798, 12.61491599534378, 13.33359344278700, 13.70066238136261, 14.11643477129164, 15.18986167297631, 15.48240765259828
