| L(s) = 1 | − 5-s − 2·13-s + 6·17-s + 19-s − 4·23-s + 25-s + 6·29-s − 2·37-s + 2·41-s − 4·43-s + 4·47-s − 7·49-s + 6·53-s − 12·59-s − 2·61-s + 2·65-s + 4·67-s − 6·73-s + 8·79-s + 8·83-s − 6·85-s − 14·89-s − 95-s + 2·97-s + 18·101-s + 20·107-s + 6·109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.554·13-s + 1.45·17-s + 0.229·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.583·47-s − 49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.702·73-s + 0.900·79-s + 0.878·83-s − 0.650·85-s − 1.48·89-s − 0.102·95-s + 0.203·97-s + 1.79·101-s + 1.93·107-s + 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669972532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669972532\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−7.86312805017825091570673097734, −7.44218819606865779004077247635, −6.58394915735401333840280937690, −5.85301552104361865058309984221, −5.09129255514460324966381486154, −4.40710329399950691308505029011, −3.50848174280177984641774825906, −2.87831481860071704058538315167, −1.77392413079994027798064751247, −0.66637807692875875172301411675,
0.66637807692875875172301411675, 1.77392413079994027798064751247, 2.87831481860071704058538315167, 3.50848174280177984641774825906, 4.40710329399950691308505029011, 5.09129255514460324966381486154, 5.85301552104361865058309984221, 6.58394915735401333840280937690, 7.44218819606865779004077247635, 7.86312805017825091570673097734
