| L(s) = 1 | + 5-s − 4·7-s − 13-s − 6·17-s + 4·19-s + 6·23-s + 25-s + 6·29-s − 10·31-s − 4·35-s + 10·37-s − 6·41-s + 4·43-s + 12·47-s + 9·49-s − 12·53-s + 12·59-s + 10·61-s − 65-s − 14·67-s − 16·73-s + 8·79-s − 12·83-s − 6·85-s + 6·89-s + 4·91-s + 4·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.277·13-s − 1.45·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 1.79·31-s − 0.676·35-s + 1.64·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 9/7·49-s − 1.64·53-s + 1.56·59-s + 1.28·61-s − 0.124·65-s − 1.71·67-s − 1.87·73-s + 0.900·79-s − 1.31·83-s − 0.650·85-s + 0.635·89-s + 0.419·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.470558285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470558285\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.73162914278970, −14.47071440343230, −13.58580057228065, −13.26163964207670, −12.92378403067985, −12.41370219328380, −11.70713524771734, −11.15615412670010, −10.58557017940589, −10.04228801479187, −9.508766566365829, −9.011142737201124, −8.779064386807625, −7.707211699594507, −7.132818071379838, −6.740873308830465, −6.155502280184557, −5.599608142189016, −4.929530338667327, −4.224442589119622, −3.532063068531879, −2.785706395219764, −2.469848702379827, −1.360254728965712, −0.4572687679455967,
0.4572687679455967, 1.360254728965712, 2.469848702379827, 2.785706395219764, 3.532063068531879, 4.224442589119622, 4.929530338667327, 5.599608142189016, 6.155502280184557, 6.740873308830465, 7.132818071379838, 7.707211699594507, 8.779064386807625, 9.011142737201124, 9.508766566365829, 10.04228801479187, 10.58557017940589, 11.15615412670010, 11.70713524771734, 12.41370219328380, 12.92378403067985, 13.26163964207670, 13.58580057228065, 14.47071440343230, 14.73162914278970
