| L(s) = 1 | − 5-s − 7-s + 11-s − 3·13-s + 2·19-s + 4·23-s − 4·25-s + 2·31-s + 35-s + 7·37-s + 11·43-s + 10·47-s + 49-s + 9·53-s − 55-s − 4·59-s − 4·61-s + 3·65-s + 13·67-s − 8·71-s + 73-s − 77-s − 79-s − 9·83-s + 3·89-s + 3·91-s − 2·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.832·13-s + 0.458·19-s + 0.834·23-s − 4/5·25-s + 0.359·31-s + 0.169·35-s + 1.15·37-s + 1.67·43-s + 1.45·47-s + 1/7·49-s + 1.23·53-s − 0.134·55-s − 0.520·59-s − 0.512·61-s + 0.372·65-s + 1.58·67-s − 0.949·71-s + 0.117·73-s − 0.113·77-s − 0.112·79-s − 0.987·83-s + 0.317·89-s + 0.314·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.494166411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.494166411\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.76816090363163, −12.22162676512536, −11.81161674221970, −11.41522359336867, −10.93065177406158, −10.41622086328272, −9.894413647576456, −9.514573197687488, −9.050611797731303, −8.645323194756203, −7.942802386892922, −7.531734933387968, −7.204558175126520, −6.716132184709804, −5.994991501905049, −5.711528424037143, −5.101492311321991, −4.415014160291208, −4.147581608268130, −3.540465536400285, −2.842753166226157, −2.530835067301171, −1.802488541671634, −0.9017571062240594, −0.5272872726980677,
0.5272872726980677, 0.9017571062240594, 1.802488541671634, 2.530835067301171, 2.842753166226157, 3.540465536400285, 4.147581608268130, 4.415014160291208, 5.101492311321991, 5.711528424037143, 5.994991501905049, 6.716132184709804, 7.204558175126520, 7.531734933387968, 7.942802386892922, 8.645323194756203, 9.050611797731303, 9.514573197687488, 9.894413647576456, 10.41622086328272, 10.93065177406158, 11.41522359336867, 11.81161674221970, 12.22162676512536, 12.76816090363163
