| L(s) = 1 | + 4·7-s − 6·11-s + 13-s − 6·17-s − 2·19-s + 6·23-s − 6·29-s + 2·31-s + 2·37-s + 6·41-s + 2·43-s − 12·47-s + 9·49-s − 6·53-s + 6·59-s − 2·61-s − 4·67-s + 6·71-s + 10·73-s − 24·77-s − 4·79-s + 6·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.80·11-s + 0.277·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s − 1.11·29-s + 0.359·31-s + 0.328·37-s + 0.937·41-s + 0.304·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s + 0.781·59-s − 0.256·61-s − 0.488·67-s + 0.712·71-s + 1.17·73-s − 2.73·77-s − 0.450·79-s + 0.635·89-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 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 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 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
−13.28948148958603, −12.91570226899834, −12.66523258643648, −11.78568474764030, −11.24793398956584, −11.06294194557865, −10.74504945765090, −10.21780917940019, −9.482117095200712, −9.056619375702383, −8.429974809323649, −8.165892141162026, −7.659852247084410, −7.272927440891322, −6.580033440000657, −6.076581738930449, −5.255225401855734, −5.127226823947519, −4.562435279102950, −4.126683138727562, −3.271181753817916, −2.635112370557552, −2.148160366356119, −1.662480337497124, −0.7885789909992404, 0,
0.7885789909992404, 1.662480337497124, 2.148160366356119, 2.635112370557552, 3.271181753817916, 4.126683138727562, 4.562435279102950, 5.127226823947519, 5.255225401855734, 6.076581738930449, 6.580033440000657, 7.272927440891322, 7.659852247084410, 8.165892141162026, 8.429974809323649, 9.056619375702383, 9.482117095200712, 10.21780917940019, 10.74504945765090, 11.06294194557865, 11.24793398956584, 11.78568474764030, 12.66523258643648, 12.91570226899834, 13.28948148958603
