| L(s) = 1 | − 4·5-s − 2·11-s + 5·13-s + 6·17-s + 4·19-s − 6·23-s + 11·25-s + 6·29-s + 7·31-s + 7·37-s + 2·41-s + 7·43-s + 2·47-s + 6·53-s + 8·55-s − 6·59-s − 9·61-s − 20·65-s + 7·67-s + 8·71-s + 10·73-s − 79-s − 14·83-s − 24·85-s + 12·89-s − 16·95-s − 15·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.603·11-s + 1.38·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s + 11/5·25-s + 1.11·29-s + 1.25·31-s + 1.15·37-s + 0.312·41-s + 1.06·43-s + 0.291·47-s + 0.824·53-s + 1.07·55-s − 0.781·59-s − 1.15·61-s − 2.48·65-s + 0.855·67-s + 0.949·71-s + 1.17·73-s − 0.112·79-s − 1.53·83-s − 2.60·85-s + 1.27·89-s − 1.64·95-s − 1.52·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.841292941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.841292941\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 9 T + p T^{2} \) | 1.61.j |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−15.61063357684705, −15.35530911304838, −14.46376200558562, −13.98647063613696, −13.54272874389933, −12.58314100165104, −12.29589319839927, −11.77841295306036, −11.27562648138786, −10.73971381654963, −10.13861922929887, −9.518113754369810, −8.620411931082966, −8.131717899223542, −7.822726844007984, −7.344859026996457, −6.428128021266504, −5.880348585376840, −5.109106607824320, −4.324947782581794, −3.862771707985219, −3.219741219783144, −2.636575872906723, −1.165906188590922, −0.6712666807496550,
0.6712666807496550, 1.165906188590922, 2.636575872906723, 3.219741219783144, 3.862771707985219, 4.324947782581794, 5.109106607824320, 5.880348585376840, 6.428128021266504, 7.344859026996457, 7.822726844007984, 8.131717899223542, 8.620411931082966, 9.518113754369810, 10.13861922929887, 10.73971381654963, 11.27562648138786, 11.77841295306036, 12.29589319839927, 12.58314100165104, 13.54272874389933, 13.98647063613696, 14.46376200558562, 15.35530911304838, 15.61063357684705
