| L(s) = 1 | − 7-s − 11-s − 4·13-s + 3·17-s + 5·19-s + 23-s + 2·29-s − 2·31-s + 5·37-s − 9·41-s − 12·43-s + 11·47-s − 6·49-s + 59-s − 5·71-s + 16·73-s + 77-s − 11·79-s + 6·83-s + 4·89-s + 4·91-s − 15·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.301·11-s − 1.10·13-s + 0.727·17-s + 1.14·19-s + 0.208·23-s + 0.371·29-s − 0.359·31-s + 0.821·37-s − 1.40·41-s − 1.82·43-s + 1.60·47-s − 6/7·49-s + 0.130·59-s − 0.593·71-s + 1.87·73-s + 0.113·77-s − 1.23·79-s + 0.658·83-s + 0.423·89-s + 0.419·91-s − 1.52·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−15.99155970266676, −15.33741085194612, −14.90122774975274, −14.34884129540941, −13.70894964826863, −13.30649470630628, −12.57197708308350, −12.11870142116833, −11.68162812781285, −10.98242622108475, −10.24321739114206, −9.809822844819823, −9.448599975379302, −8.607381942669825, −7.990249239214786, −7.413961612926133, −6.906710008337749, −6.226808101319059, −5.293551831547688, −5.144483897491621, −4.216680072173607, −3.332568252800238, −2.903755241471516, −2.011588057832636, −1.045258695670476, 0,
1.045258695670476, 2.011588057832636, 2.903755241471516, 3.332568252800238, 4.216680072173607, 5.144483897491621, 5.293551831547688, 6.226808101319059, 6.906710008337749, 7.413961612926133, 7.990249239214786, 8.607381942669825, 9.448599975379302, 9.809822844819823, 10.24321739114206, 10.98242622108475, 11.68162812781285, 12.11870142116833, 12.57197708308350, 13.30649470630628, 13.70894964826863, 14.34884129540941, 14.90122774975274, 15.33741085194612, 15.99155970266676
