| L(s) = 1 | − 3·9-s − 13-s − 2·17-s + 8·19-s − 4·23-s − 2·29-s + 4·31-s − 6·37-s + 10·41-s + 8·47-s − 7·49-s − 6·53-s − 8·59-s − 2·61-s + 4·67-s + 12·71-s − 10·73-s + 8·79-s + 9·81-s + 12·83-s + 10·89-s + 14·97-s + 6·101-s − 4·103-s − 2·109-s + 14·113-s + 3·117-s + ⋯ |
| L(s) = 1 | − 9-s − 0.277·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s − 0.371·29-s + 0.718·31-s − 0.986·37-s + 1.56·41-s + 1.16·47-s − 49-s − 0.824·53-s − 1.04·59-s − 0.256·61-s + 0.488·67-s + 1.42·71-s − 1.17·73-s + 0.900·79-s + 81-s + 1.31·83-s + 1.05·89-s + 1.42·97-s + 0.597·101-s − 0.394·103-s − 0.191·109-s + 1.31·113-s + 0.277·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.545651682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545651682\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.982749273052339469034771965810, −7.72444694198423313772384902870, −6.72939511465811026375278353589, −5.99428347419393793443517325098, −5.35082231981171495133375129904, −4.61943326777515450117545179254, −3.58104309113781690448462556541, −2.89542969562726110443794165376, −1.97261591183835729504137621519, −0.65931071985509195594949640977,
0.65931071985509195594949640977, 1.97261591183835729504137621519, 2.89542969562726110443794165376, 3.58104309113781690448462556541, 4.61943326777515450117545179254, 5.35082231981171495133375129904, 5.99428347419393793443517325098, 6.72939511465811026375278353589, 7.72444694198423313772384902870, 7.982749273052339469034771965810
