| L(s) = 1 | + 3-s + 3·5-s + 7-s + 9-s + 5·11-s + 6·13-s + 3·15-s − 5·17-s − 19-s + 21-s + 4·23-s + 4·25-s + 27-s − 6·29-s + 6·31-s + 5·33-s + 3·35-s + 8·37-s + 6·39-s − 8·41-s − 9·43-s + 3·45-s + 47-s − 6·49-s − 5·51-s − 2·53-s + 15·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.66·13-s + 0.774·15-s − 1.21·17-s − 0.229·19-s + 0.218·21-s + 0.834·23-s + 4/5·25-s + 0.192·27-s − 1.11·29-s + 1.07·31-s + 0.870·33-s + 0.507·35-s + 1.31·37-s + 0.960·39-s − 1.24·41-s − 1.37·43-s + 0.447·45-s + 0.145·47-s − 6/7·49-s − 0.700·51-s − 0.274·53-s + 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.811452923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.811452923\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−8.791997865595783924823926678815, −8.009694737499873447589050746513, −6.70133446939976904656819214588, −6.49749160169048802133545940441, −5.69112412235536456725087217091, −4.63178168759724390480509005391, −3.89058231761553340258241026684, −2.94072738324304931925847753623, −1.79010635851573294160738022151, −1.33686447944871733441234003392,
1.33686447944871733441234003392, 1.79010635851573294160738022151, 2.94072738324304931925847753623, 3.89058231761553340258241026684, 4.63178168759724390480509005391, 5.69112412235536456725087217091, 6.49749160169048802133545940441, 6.70133446939976904656819214588, 8.009694737499873447589050746513, 8.791997865595783924823926678815
