| L(s) = 1 | − 3·9-s − 5·11-s − 6·13-s + 4·17-s + 6·19-s − 3·23-s − 3·29-s − 2·31-s − 7·37-s + 4·41-s + 7·43-s − 2·47-s + 10·53-s + 14·59-s − 4·61-s − 3·67-s − 13·71-s + 16·73-s + 79-s + 9·81-s + 10·83-s − 10·89-s − 2·97-s + 15·99-s − 12·101-s + 2·103-s + 12·107-s + ⋯ |
| L(s) = 1 | − 9-s − 1.50·11-s − 1.66·13-s + 0.970·17-s + 1.37·19-s − 0.625·23-s − 0.557·29-s − 0.359·31-s − 1.15·37-s + 0.624·41-s + 1.06·43-s − 0.291·47-s + 1.37·53-s + 1.82·59-s − 0.512·61-s − 0.366·67-s − 1.54·71-s + 1.87·73-s + 0.112·79-s + 81-s + 1.09·83-s − 1.05·89-s − 0.203·97-s + 1.50·99-s − 1.19·101-s + 0.197·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067690315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067690315\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.085599298762783013124091966345, −7.56585980259470957339745405964, −7.10254123384146647615715096321, −5.68501340342797051597967594604, −5.50090383178162094404309343694, −4.79562655167183435898050199796, −3.56278934217657163230544922405, −2.80046475792476367629586590343, −2.15847618483313600210258262059, −0.53302952297629341170769605048,
0.53302952297629341170769605048, 2.15847618483313600210258262059, 2.80046475792476367629586590343, 3.56278934217657163230544922405, 4.79562655167183435898050199796, 5.50090383178162094404309343694, 5.68501340342797051597967594604, 7.10254123384146647615715096321, 7.56585980259470957339745405964, 8.085599298762783013124091966345
