| L(s) = 1 | + 5-s + 6·11-s + 6·13-s − 4·17-s + 2·19-s + 8·23-s + 25-s − 29-s − 2·31-s + 2·41-s − 7·49-s + 14·53-s + 6·55-s − 4·59-s + 6·61-s + 6·65-s − 12·67-s + 12·71-s − 16·73-s − 2·79-s − 12·83-s − 4·85-s + 2·89-s + 2·95-s − 12·97-s − 18·101-s + 16·103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.80·11-s + 1.66·13-s − 0.970·17-s + 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.185·29-s − 0.359·31-s + 0.312·41-s − 49-s + 1.92·53-s + 0.809·55-s − 0.520·59-s + 0.768·61-s + 0.744·65-s − 1.46·67-s + 1.42·71-s − 1.87·73-s − 0.225·79-s − 1.31·83-s − 0.433·85-s + 0.211·89-s + 0.205·95-s − 1.21·97-s − 1.79·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.836721568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.836721568\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 29 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−8.513077455812255446528416710331, −7.27789919764118822706359733344, −6.70607134688902911171124468494, −6.15654826081556801683727372401, −5.41455840500070055565313996269, −4.38185690418490497827832247613, −3.76887481885124968449186648583, −2.92888751042091936492370199975, −1.65926544369794670209185082702, −1.02787257655212437693741336580,
1.02787257655212437693741336580, 1.65926544369794670209185082702, 2.92888751042091936492370199975, 3.76887481885124968449186648583, 4.38185690418490497827832247613, 5.41455840500070055565313996269, 6.15654826081556801683727372401, 6.70607134688902911171124468494, 7.27789919764118822706359733344, 8.513077455812255446528416710331
