| L(s) = 1 | − 5-s − 7-s − 4·11-s + 13-s + 7·17-s − 19-s + 5·23-s + 25-s − 7·29-s − 2·31-s + 35-s − 6·37-s − 6·41-s + 10·43-s + 8·47-s − 6·49-s + 3·53-s + 4·55-s − 5·59-s − 8·61-s − 65-s + 11·67-s + 12·71-s − 9·73-s + 4·77-s + 6·79-s − 14·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.20·11-s + 0.277·13-s + 1.69·17-s − 0.229·19-s + 1.04·23-s + 1/5·25-s − 1.29·29-s − 0.359·31-s + 0.169·35-s − 0.986·37-s − 0.937·41-s + 1.52·43-s + 1.16·47-s − 6/7·49-s + 0.412·53-s + 0.539·55-s − 0.650·59-s − 1.02·61-s − 0.124·65-s + 1.34·67-s + 1.42·71-s − 1.05·73-s + 0.455·77-s + 0.675·79-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.51784017025422563991017652671, −7.18843357090761635673368072709, −6.11121317054942529327319722129, −5.45034639407982918561832958173, −4.91356602145245611087684659901, −3.77115849210867314024238891079, −3.28041305811333282223194530425, −2.40486544508861371430359964799, −1.19087560489374809732073250999, 0,
1.19087560489374809732073250999, 2.40486544508861371430359964799, 3.28041305811333282223194530425, 3.77115849210867314024238891079, 4.91356602145245611087684659901, 5.45034639407982918561832958173, 6.11121317054942529327319722129, 7.18843357090761635673368072709, 7.51784017025422563991017652671
