| L(s) = 1 | + 2·3-s − 5-s + 4·7-s + 9-s − 4·13-s − 2·15-s − 5·17-s − 2·19-s + 8·21-s + 6·23-s − 4·25-s − 4·27-s − 6·29-s + 4·31-s − 4·35-s − 37-s − 8·39-s − 12·41-s − 2·43-s − 45-s + 7·47-s + 9·49-s − 10·51-s − 2·53-s − 4·57-s + 14·61-s + 4·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s − 1.21·17-s − 0.458·19-s + 1.74·21-s + 1.25·23-s − 4/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.676·35-s − 0.164·37-s − 1.28·39-s − 1.87·41-s − 0.304·43-s − 0.149·45-s + 1.02·47-s + 9/7·49-s − 1.40·51-s − 0.274·53-s − 0.529·57-s + 1.79·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.779545209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.779545209\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.13798130257242, −13.82446043878719, −13.08791042028859, −12.86827808352830, −11.95996542330875, −11.54209441264853, −11.26156511966055, −10.64628848325585, −9.977983276943009, −9.455228199871586, −8.822970400545700, −8.461353897764643, −8.130716642965996, −7.482996900830074, −7.134299055870369, −6.537188117013567, −5.482004549748334, −5.126810222666280, −4.472000384766768, −4.034158893907911, −3.350826077146185, −2.584199606861873, −2.096778067133286, −1.648344774587883, −0.4812153439458405,
0.4812153439458405, 1.648344774587883, 2.096778067133286, 2.584199606861873, 3.350826077146185, 4.034158893907911, 4.472000384766768, 5.126810222666280, 5.482004549748334, 6.537188117013567, 7.134299055870369, 7.482996900830074, 8.130716642965996, 8.461353897764643, 8.822970400545700, 9.455228199871586, 9.977983276943009, 10.64628848325585, 11.26156511966055, 11.54209441264853, 11.95996542330875, 12.86827808352830, 13.08791042028859, 13.82446043878719, 14.13798130257242
