| L(s) = 1 | + 3-s − 3·7-s + 9-s − 2·11-s + 5·13-s + 17-s − 7·19-s − 3·21-s + 4·23-s + 27-s − 4·29-s + 5·31-s − 2·33-s − 6·37-s + 5·39-s + 2·41-s − 5·43-s − 12·47-s + 2·49-s + 51-s − 6·53-s − 7·57-s + 10·59-s + 13·61-s − 3·63-s − 5·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.603·11-s + 1.38·13-s + 0.242·17-s − 1.60·19-s − 0.654·21-s + 0.834·23-s + 0.192·27-s − 0.742·29-s + 0.898·31-s − 0.348·33-s − 0.986·37-s + 0.800·39-s + 0.312·41-s − 0.762·43-s − 1.75·47-s + 2/7·49-s + 0.140·51-s − 0.824·53-s − 0.927·57-s + 1.30·59-s + 1.66·61-s − 0.377·63-s − 0.610·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−8.137184218715327838873003326071, −6.99412819668853863894480711178, −6.53829584423759425873693922697, −5.83474504309947961924395003429, −4.87583797334088801429137897462, −3.88873162191321579918528201636, −3.32808273731244561883285330635, −2.53540868354575814051311845420, −1.44134124524462685323064662958, 0,
1.44134124524462685323064662958, 2.53540868354575814051311845420, 3.32808273731244561883285330635, 3.88873162191321579918528201636, 4.87583797334088801429137897462, 5.83474504309947961924395003429, 6.53829584423759425873693922697, 6.99412819668853863894480711178, 8.137184218715327838873003326071
