| L(s) = 1 | + 3-s − 2·7-s + 9-s − 13-s − 2·17-s + 2·19-s − 2·21-s + 8·23-s − 5·25-s + 27-s − 6·29-s + 2·31-s − 6·37-s − 39-s − 4·43-s + 8·47-s − 3·49-s − 2·51-s − 6·53-s + 2·57-s + 4·59-s − 2·61-s − 2·63-s + 2·67-s + 8·69-s + 4·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.277·13-s − 0.485·17-s + 0.458·19-s − 0.436·21-s + 1.66·23-s − 25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.986·37-s − 0.160·39-s − 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s + 0.264·57-s + 0.520·59-s − 0.256·61-s − 0.251·63-s + 0.244·67-s + 0.963·69-s + 0.474·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.59391030882500, −13.06387724816158, −12.82254762381544, −12.16034080170099, −11.76481400729918, −11.06367146853700, −10.80141865632432, −10.05009171766168, −9.674998317648744, −9.286015582432235, −8.796819997498179, −8.357065421427173, −7.575525733989758, −7.352789347143053, −6.728135107795259, −6.334686162933251, −5.588474847019796, −5.135905428320136, −4.534410897727821, −3.860768419453049, −3.356393332876313, −2.945859832900198, −2.225928021254197, −1.680511971981524, −0.8153634369041349, 0,
0.8153634369041349, 1.680511971981524, 2.225928021254197, 2.945859832900198, 3.356393332876313, 3.860768419453049, 4.534410897727821, 5.135905428320136, 5.588474847019796, 6.334686162933251, 6.728135107795259, 7.352789347143053, 7.575525733989758, 8.357065421427173, 8.796819997498179, 9.286015582432235, 9.674998317648744, 10.05009171766168, 10.80141865632432, 11.06367146853700, 11.76481400729918, 12.16034080170099, 12.82254762381544, 13.06387724816158, 13.59391030882500
