| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 3·11-s − 2·13-s − 2·14-s + 16-s − 3·17-s − 19-s + 3·22-s − 6·23-s − 2·26-s − 2·28-s − 6·29-s − 4·31-s + 32-s − 3·34-s + 4·37-s − 38-s − 9·41-s + 43-s + 3·44-s − 6·46-s − 6·47-s − 3·49-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.904·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s − 0.229·19-s + 0.639·22-s − 1.25·23-s − 0.392·26-s − 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.657·37-s − 0.162·38-s − 1.40·41-s + 0.152·43-s + 0.452·44-s − 0.884·46-s − 0.875·47-s − 3/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−7.969576123757136118804430099135, −7.08177789320983458641691824871, −6.56600580052840486429940471850, −5.88646034593949749002861665106, −5.07098129826932849586856478197, −4.08330306314885901878996878220, −3.64220858429912662305662465254, −2.55410427597476064388867668021, −1.68561011358576029662885717150, 0,
1.68561011358576029662885717150, 2.55410427597476064388867668021, 3.64220858429912662305662465254, 4.08330306314885901878996878220, 5.07098129826932849586856478197, 5.88646034593949749002861665106, 6.56600580052840486429940471850, 7.08177789320983458641691824871, 7.969576123757136118804430099135
