| L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s − 2·7-s + 8-s + 9-s − 4·10-s + 12-s − 2·14-s − 4·15-s + 16-s + 18-s − 4·19-s − 4·20-s − 2·21-s − 6·23-s + 24-s + 11·25-s + 27-s − 2·28-s + 4·29-s − 4·30-s − 6·31-s + 32-s + 8·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.288·12-s − 0.534·14-s − 1.03·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.894·20-s − 0.436·21-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s − 0.377·28-s + 0.742·29-s − 0.730·30-s − 1.07·31-s + 0.176·32-s + 1.35·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.99696029620825, −12.46524608349546, −12.06908679219005, −11.74269316963866, −11.24910545630248, −10.68276059502044, −10.30284926986902, −9.807597937154376, −9.163206587571743, −8.546785819185665, −8.299105598257687, −7.831285276977191, −7.332915465062032, −6.830405677614963, −6.491752020241855, −5.948932759305612, −5.112167558397070, −4.693966197001986, −4.190833513687216, −3.729892653420926, −3.236912292678465, −3.115925599115309, −2.098836112200821, −1.708594595180982, −0.5884830559967882, 0,
0.5884830559967882, 1.708594595180982, 2.098836112200821, 3.115925599115309, 3.236912292678465, 3.729892653420926, 4.190833513687216, 4.693966197001986, 5.112167558397070, 5.948932759305612, 6.491752020241855, 6.830405677614963, 7.332915465062032, 7.831285276977191, 8.299105598257687, 8.546785819185665, 9.163206587571743, 9.807597937154376, 10.30284926986902, 10.68276059502044, 11.24910545630248, 11.74269316963866, 12.06908679219005, 12.46524608349546, 12.99696029620825
