| L(s) = 1 | − 2·4-s + 3·11-s + 4·13-s + 4·16-s + 7·17-s + 3·19-s − 5·23-s − 29-s + 31-s + 8·41-s + 10·43-s − 6·44-s + 2·47-s − 7·49-s − 8·52-s − 9·53-s + 12·59-s − 2·61-s − 8·64-s + 11·67-s − 14·68-s + 2·71-s − 10·73-s − 6·76-s − 10·79-s + 3·83-s − 7·89-s + ⋯ |
| L(s) = 1 | − 4-s + 0.904·11-s + 1.10·13-s + 16-s + 1.69·17-s + 0.688·19-s − 1.04·23-s − 0.185·29-s + 0.179·31-s + 1.24·41-s + 1.52·43-s − 0.904·44-s + 0.291·47-s − 49-s − 1.10·52-s − 1.23·53-s + 1.56·59-s − 0.256·61-s − 64-s + 1.34·67-s − 1.69·68-s + 0.237·71-s − 1.17·73-s − 0.688·76-s − 1.12·79-s + 0.329·83-s − 0.741·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.970043234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970043234\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−7.953130262237492561476243466600, −7.49532034420196734748879731989, −6.35030469901405090326754540607, −5.80857599496988502553767807658, −5.20453864467659136189984011924, −4.12557734124254618073263874531, −3.79354947968001088836524721550, −2.95597753122059900766596203721, −1.48683397358702736784526648726, −0.812494885330245873155671424791,
0.812494885330245873155671424791, 1.48683397358702736784526648726, 2.95597753122059900766596203721, 3.79354947968001088836524721550, 4.12557734124254618073263874531, 5.20453864467659136189984011924, 5.80857599496988502553767807658, 6.35030469901405090326754540607, 7.49532034420196734748879731989, 7.953130262237492561476243466600
