| L(s) = 1 | − 11-s − 4·17-s − 4·19-s + 6·23-s − 2·29-s + 6·37-s + 10·41-s − 4·43-s + 10·47-s − 7·49-s + 2·53-s + 4·59-s − 14·61-s − 2·67-s − 4·71-s + 4·73-s − 8·79-s + 12·83-s − 6·89-s − 6·97-s − 14·101-s − 10·103-s + 8·107-s + 14·109-s + 14·113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s − 0.970·17-s − 0.917·19-s + 1.25·23-s − 0.371·29-s + 0.986·37-s + 1.56·41-s − 0.609·43-s + 1.45·47-s − 49-s + 0.274·53-s + 0.520·59-s − 1.79·61-s − 0.244·67-s − 0.474·71-s + 0.468·73-s − 0.900·79-s + 1.31·83-s − 0.635·89-s − 0.609·97-s − 1.39·101-s − 0.985·103-s + 0.773·107-s + 1.34·109-s + 1.31·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−7.38575097138221563854172013957, −6.58121592543446870435854148235, −6.07393799041159709884848839464, −5.22056940312893237540509804739, −4.51933912115413931783911319027, −3.92815071409613961902476484743, −2.86668402403538005827985605128, −2.29508011196570541344899599114, −1.19804820094616576458048306914, 0,
1.19804820094616576458048306914, 2.29508011196570541344899599114, 2.86668402403538005827985605128, 3.92815071409613961902476484743, 4.51933912115413931783911319027, 5.22056940312893237540509804739, 6.07393799041159709884848839464, 6.58121592543446870435854148235, 7.38575097138221563854172013957
