| L(s) = 1 | − 5-s − 4·7-s − 5·13-s − 4·17-s + 19-s + 4·23-s − 4·25-s + 10·29-s − 4·31-s + 4·35-s + 2·37-s − 8·41-s + 43-s + 4·47-s + 9·49-s − 53-s + 8·59-s − 14·61-s + 5·65-s − 12·67-s + 71-s − 2·73-s − 8·79-s + 83-s + 4·85-s + 12·89-s + 20·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 1.38·13-s − 0.970·17-s + 0.229·19-s + 0.834·23-s − 4/5·25-s + 1.85·29-s − 0.718·31-s + 0.676·35-s + 0.328·37-s − 1.24·41-s + 0.152·43-s + 0.583·47-s + 9/7·49-s − 0.137·53-s + 1.04·59-s − 1.79·61-s + 0.620·65-s − 1.46·67-s + 0.118·71-s − 0.234·73-s − 0.900·79-s + 0.109·83-s + 0.433·85-s + 1.27·89-s + 2.09·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−13.10590590409797, −12.72201087515636, −12.16739677938659, −11.93264536981264, −11.44138419606899, −10.69950817411768, −10.35452046150382, −9.891139104397661, −9.474562667354358, −8.953961431483347, −8.643318296668396, −7.858348697761972, −7.382980250984263, −7.030874648481698, −6.484963000428870, −6.159672532596728, −5.438011738651095, −4.845564957276843, −4.446925309357370, −3.816645646548547, −3.219479591143512, −2.771160109705151, −2.333067477634633, −1.458328889788819, −0.5210064548620432, 0,
0.5210064548620432, 1.458328889788819, 2.333067477634633, 2.771160109705151, 3.219479591143512, 3.816645646548547, 4.446925309357370, 4.845564957276843, 5.438011738651095, 6.159672532596728, 6.484963000428870, 7.030874648481698, 7.382980250984263, 7.858348697761972, 8.643318296668396, 8.953961431483347, 9.474562667354358, 9.891139104397661, 10.35452046150382, 10.69950817411768, 11.44138419606899, 11.93264536981264, 12.16739677938659, 12.72201087515636, 13.10590590409797
