| L(s) = 1 | + 2·2-s + 2·4-s + 3·5-s − 5·7-s + 6·10-s − 11-s + 2·13-s − 10·14-s − 4·16-s + 17-s − 19-s + 6·20-s − 2·22-s + 4·23-s + 4·25-s + 4·26-s − 10·28-s + 2·29-s − 6·31-s − 8·32-s + 2·34-s − 15·35-s − 2·38-s − 43-s − 2·44-s + 8·46-s + 9·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.34·5-s − 1.88·7-s + 1.89·10-s − 0.301·11-s + 0.554·13-s − 2.67·14-s − 16-s + 0.242·17-s − 0.229·19-s + 1.34·20-s − 0.426·22-s + 0.834·23-s + 4/5·25-s + 0.784·26-s − 1.88·28-s + 0.371·29-s − 1.07·31-s − 1.41·32-s + 0.342·34-s − 2.53·35-s − 0.324·38-s − 0.152·43-s − 0.301·44-s + 1.17·46-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.215096110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.215096110\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.00980511664590576338005674082, −12.37438605696103795680934752253, −10.84385583714429238920836096770, −9.769325574936890193507770039971, −9.028755790922139382896168795846, −6.85073886350288365363080648917, −6.12241751609850681792173625996, −5.34668146252658900228619106104, −3.67840796781373175574076394185, −2.61687998381550726785939862479,
2.61687998381550726785939862479, 3.67840796781373175574076394185, 5.34668146252658900228619106104, 6.12241751609850681792173625996, 6.85073886350288365363080648917, 9.028755790922139382896168795846, 9.769325574936890193507770039971, 10.84385583714429238920836096770, 12.37438605696103795680934752253, 13.00980511664590576338005674082
