| L(s) = 1 | − 2·7-s − 3·11-s − 4·13-s + 3·17-s − 5·19-s + 6·23-s − 2·31-s + 2·37-s + 3·41-s + 4·43-s + 12·47-s − 3·49-s − 6·53-s + 2·61-s + 13·67-s + 12·71-s + 11·73-s + 6·77-s + 10·79-s − 9·83-s − 15·89-s + 8·91-s + 2·97-s + 18·101-s + 4·103-s − 3·107-s − 10·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.904·11-s − 1.10·13-s + 0.727·17-s − 1.14·19-s + 1.25·23-s − 0.359·31-s + 0.328·37-s + 0.468·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.256·61-s + 1.58·67-s + 1.42·71-s + 1.28·73-s + 0.683·77-s + 1.12·79-s − 0.987·83-s − 1.58·89-s + 0.838·91-s + 0.203·97-s + 1.79·101-s + 0.394·103-s − 0.290·107-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.235237982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.235237982\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.525394861615405396000845166993, −7.74259453263933502686277981111, −7.12027035410652945790309606799, −6.36901426807204320186985434377, −5.48177790941829296116695062663, −4.85247836211906827996531873036, −3.85823879659342768603780553916, −2.89341560511568671579379393600, −2.23294311234448454772728765221, −0.62279377578968419349957990411,
0.62279377578968419349957990411, 2.23294311234448454772728765221, 2.89341560511568671579379393600, 3.85823879659342768603780553916, 4.85247836211906827996531873036, 5.48177790941829296116695062663, 6.36901426807204320186985434377, 7.12027035410652945790309606799, 7.74259453263933502686277981111, 8.525394861615405396000845166993
