| L(s) = 1 | − 2·7-s + 5·11-s + 13-s + 6·17-s + 4·23-s − 5·25-s − 4·29-s + 6·31-s − 4·37-s + 11·41-s + 12·43-s + 10·47-s − 3·49-s + 12·53-s + 11·59-s − 2·61-s + 3·67-s + 6·71-s − 3·73-s − 10·77-s − 2·79-s + 9·83-s − 10·89-s − 2·91-s − 7·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.50·11-s + 0.277·13-s + 1.45·17-s + 0.834·23-s − 25-s − 0.742·29-s + 1.07·31-s − 0.657·37-s + 1.71·41-s + 1.82·43-s + 1.45·47-s − 3/7·49-s + 1.64·53-s + 1.43·59-s − 0.256·61-s + 0.366·67-s + 0.712·71-s − 0.351·73-s − 1.13·77-s − 0.225·79-s + 0.987·83-s − 1.05·89-s − 0.209·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 337896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 337896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.127761386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.127761386\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.46839091365008, −12.16848072637222, −11.78468787546329, −11.19420406410136, −10.86176450853579, −10.18884703746848, −9.790914295578426, −9.413461461125137, −9.007041863331929, −8.591348940596304, −7.884290056823513, −7.488767379822340, −6.977641051281692, −6.583921103816665, −5.889929935231920, −5.751280822734256, −5.180381286648458, −4.229829701874666, −3.975289358445905, −3.620483304876198, −2.882464916789614, −2.456167284375521, −1.635065012713359, −0.9411569463751097, −0.6742575083924847,
0.6742575083924847, 0.9411569463751097, 1.635065012713359, 2.456167284375521, 2.882464916789614, 3.620483304876198, 3.975289358445905, 4.229829701874666, 5.180381286648458, 5.751280822734256, 5.889929935231920, 6.583921103816665, 6.977641051281692, 7.488767379822340, 7.884290056823513, 8.591348940596304, 9.007041863331929, 9.413461461125137, 9.790914295578426, 10.18884703746848, 10.86176450853579, 11.19420406410136, 11.78468787546329, 12.16848072637222, 12.46839091365008
