| L(s) = 1 | − 5-s − 2·7-s − 3·9-s − 11-s − 7·17-s − 7·19-s − 4·23-s + 25-s + 6·29-s − 8·31-s + 2·35-s + 7·37-s + 3·41-s + 9·43-s + 3·45-s − 7·47-s − 3·49-s − 6·53-s + 55-s − 14·59-s + 2·61-s + 6·63-s + 5·67-s − 2·71-s + 8·73-s + 2·77-s − 4·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 9-s − 0.301·11-s − 1.69·17-s − 1.60·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.338·35-s + 1.15·37-s + 0.468·41-s + 1.37·43-s + 0.447·45-s − 1.02·47-s − 3/7·49-s − 0.824·53-s + 0.134·55-s − 1.82·59-s + 0.256·61-s + 0.755·63-s + 0.610·67-s − 0.237·71-s + 0.936·73-s + 0.227·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−13.46219487727799, −13.06831810031967, −12.66200503798000, −12.27061376953114, −11.62795929524520, −11.07280908625813, −10.82494302913233, −10.45836881981566, −9.510625970775927, −9.313533100812875, −8.772089067605339, −8.151129522548128, −7.988480829973709, −7.185856179714725, −6.537542698550419, −6.269840836136953, −5.876723443402762, −5.016829883495142, −4.515205985943464, −4.035949732902144, −3.458647647328701, −2.684948462562433, −2.417890853507327, −1.679996023333398, −0.4737061976656310, 0,
0.4737061976656310, 1.679996023333398, 2.417890853507327, 2.684948462562433, 3.458647647328701, 4.035949732902144, 4.515205985943464, 5.016829883495142, 5.876723443402762, 6.269840836136953, 6.537542698550419, 7.185856179714725, 7.988480829973709, 8.151129522548128, 8.772089067605339, 9.313533100812875, 9.510625970775927, 10.45836881981566, 10.82494302913233, 11.07280908625813, 11.62795929524520, 12.27061376953114, 12.66200503798000, 13.06831810031967, 13.46219487727799
