| L(s) = 1 | − 11-s + 2·13-s + 4·17-s + 6·19-s + 4·23-s − 5·25-s + 2·29-s − 10·31-s − 6·37-s − 12·41-s − 12·43-s − 6·47-s + 6·53-s − 6·61-s − 4·67-s + 8·71-s − 8·73-s + 16·79-s − 2·83-s + 10·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 0.554·13-s + 0.970·17-s + 1.37·19-s + 0.834·23-s − 25-s + 0.371·29-s − 1.79·31-s − 0.986·37-s − 1.87·41-s − 1.82·43-s − 0.875·47-s + 0.824·53-s − 0.768·61-s − 0.488·67-s + 0.949·71-s − 0.936·73-s + 1.80·79-s − 0.219·83-s + 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.56949686430649, −13.21931496976516, −12.56419255974749, −11.99498360288769, −11.75264624335927, −11.21688090133414, −10.66480344735900, −10.13008833358434, −9.812069828680642, −9.250618222792423, −8.660883464237994, −8.307338041178889, −7.596965324602029, −7.326808525902313, −6.753279322024236, −6.122332704526369, −5.573751436558216, −5.058052683773652, −4.835228571754620, −3.670968691411466, −3.440892170654534, −3.103617828952209, −1.986400526218117, −1.637625801044069, −0.8546633215448874, 0,
0.8546633215448874, 1.637625801044069, 1.986400526218117, 3.103617828952209, 3.440892170654534, 3.670968691411466, 4.835228571754620, 5.058052683773652, 5.573751436558216, 6.122332704526369, 6.753279322024236, 7.326808525902313, 7.596965324602029, 8.307338041178889, 8.660883464237994, 9.250618222792423, 9.812069828680642, 10.13008833358434, 10.66480344735900, 11.21688090133414, 11.75264624335927, 11.99498360288769, 12.56419255974749, 13.21931496976516, 13.56949686430649
