| L(s) = 1 | + 5-s + 2·7-s + 4·11-s − 4·17-s + 2·19-s − 8·23-s + 25-s − 2·31-s + 2·35-s + 10·37-s + 10·41-s − 3·49-s − 8·53-s + 4·55-s − 4·59-s + 2·61-s + 2·67-s − 2·73-s + 8·77-s + 12·83-s − 4·85-s + 6·89-s + 2·95-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 1.20·11-s − 0.970·17-s + 0.458·19-s − 1.66·23-s + 1/5·25-s − 0.359·31-s + 0.338·35-s + 1.64·37-s + 1.56·41-s − 3/7·49-s − 1.09·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.244·67-s − 0.234·73-s + 0.911·77-s + 1.31·83-s − 0.433·85-s + 0.635·89-s + 0.205·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.586025819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.586025819\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.07829963900993, −12.23701993546478, −12.01403167787735, −11.41707303182991, −11.08860887576702, −10.71989525561841, −9.918468325245261, −9.650236720598305, −9.145447575868613, −8.787069197563926, −8.125447164146523, −7.705615923557169, −7.330574439193861, −6.445687492900549, −6.278183057709764, −5.850462102382407, −5.100610866087620, −4.586908756732550, −4.176316998592709, −3.689630252401801, −2.948640533216407, −2.199502996817261, −1.879981893681283, −1.207840671950865, −0.5357633358427717,
0.5357633358427717, 1.207840671950865, 1.879981893681283, 2.199502996817261, 2.948640533216407, 3.689630252401801, 4.176316998592709, 4.586908756732550, 5.100610866087620, 5.850462102382407, 6.278183057709764, 6.445687492900549, 7.330574439193861, 7.705615923557169, 8.125447164146523, 8.787069197563926, 9.145447575868613, 9.650236720598305, 9.918468325245261, 10.71989525561841, 11.08860887576702, 11.41707303182991, 12.01403167787735, 12.23701993546478, 13.07829963900993
