| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 14-s + 15-s + 16-s + 3·17-s + 18-s + 5·19-s + 20-s − 21-s + 22-s − 6·23-s + 24-s − 4·25-s + 27-s − 28-s + 30-s − 6·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 1.14·19-s + 0.223·20-s − 0.218·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.188·28-s + 0.182·30-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.50188298792830, −14.14107481069118, −13.65937312523144, −13.30676668761170, −12.68635294914114, −12.14559431095276, −11.83757963897568, −11.12489427160672, −10.55341348993222, −9.923039236127133, −9.461832537857018, −9.251902551095845, −8.218963811986051, −7.791698079121002, −7.412966980425751, −6.543652576123614, −6.221996169281996, −5.486428894156101, −5.159534868643593, −4.165822202560612, −3.830577910949337, −3.125382114394542, −2.660865366838596, −1.751917467630049, −1.341009702203164, 0,
1.341009702203164, 1.751917467630049, 2.660865366838596, 3.125382114394542, 3.830577910949337, 4.165822202560612, 5.159534868643593, 5.486428894156101, 6.221996169281996, 6.543652576123614, 7.412966980425751, 7.791698079121002, 8.218963811986051, 9.251902551095845, 9.461832537857018, 9.923039236127133, 10.55341348993222, 11.12489427160672, 11.83757963897568, 12.14559431095276, 12.68635294914114, 13.30676668761170, 13.65937312523144, 14.14107481069118, 14.50188298792830
