| L(s) = 1 | + 2·5-s − 3·11-s + 2·13-s − 7·17-s + 4·19-s − 23-s − 25-s + 9·29-s − 10·31-s + 6·37-s − 2·41-s − 2·43-s + 3·47-s + 12·53-s − 6·55-s − 12·59-s + 10·61-s + 4·65-s + 8·67-s − 3·71-s + 11·73-s + 79-s + 12·83-s − 14·85-s − 6·89-s + 8·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.904·11-s + 0.554·13-s − 1.69·17-s + 0.917·19-s − 0.208·23-s − 1/5·25-s + 1.67·29-s − 1.79·31-s + 0.986·37-s − 0.312·41-s − 0.304·43-s + 0.437·47-s + 1.64·53-s − 0.809·55-s − 1.56·59-s + 1.28·61-s + 0.496·65-s + 0.977·67-s − 0.356·71-s + 1.28·73-s + 0.112·79-s + 1.31·83-s − 1.51·85-s − 0.635·89-s + 0.820·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.95134621029434, −13.71666653499266, −13.34740188304568, −12.87946655532703, −12.32355918684357, −11.70995850020872, −11.13634672358914, −10.70684245924936, −10.32513877779204, −9.575604497567039, −9.360271263149959, −8.679042008571759, −8.211869943809745, −7.654543134924664, −6.931853980020443, −6.549289992760060, −5.951472739429139, −5.342463887649643, −5.049136225662844, −4.190025154229369, −3.721742107578280, −2.763405831873241, −2.413654857483476, −1.755352882684947, −0.9470493635316955, 0,
0.9470493635316955, 1.755352882684947, 2.413654857483476, 2.763405831873241, 3.721742107578280, 4.190025154229369, 5.049136225662844, 5.342463887649643, 5.951472739429139, 6.549289992760060, 6.931853980020443, 7.654543134924664, 8.211869943809745, 8.679042008571759, 9.360271263149959, 9.575604497567039, 10.32513877779204, 10.70684245924936, 11.13634672358914, 11.70995850020872, 12.32355918684357, 12.87946655532703, 13.34740188304568, 13.71666653499266, 13.95134621029434
