| L(s) = 1 | − 3-s − 5-s + 9-s − 11-s + 2·13-s + 15-s − 3·17-s + 6·19-s − 5·23-s − 4·25-s − 27-s + 6·29-s − 4·31-s + 33-s + 2·37-s − 2·39-s + 3·41-s + 2·43-s − 45-s + 7·47-s + 3·51-s + 2·53-s + 55-s − 6·57-s − 8·59-s − 7·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 0.727·17-s + 1.37·19-s − 1.04·23-s − 4/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s + 0.468·41-s + 0.304·43-s − 0.149·45-s + 1.02·47-s + 0.420·51-s + 0.274·53-s + 0.134·55-s − 0.794·57-s − 1.04·59-s − 0.896·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.99057655542971, −13.52957409590408, −12.91818577976065, −12.39134954867641, −11.99439138492374, −11.55337585922759, −10.98710244665513, −10.75249877646945, −9.996972590913863, −9.611830513794850, −9.091144897038380, −8.383157733077394, −7.952145802025208, −7.473881381258904, −6.936746840689345, −6.326369505190371, −5.827847407965788, −5.358520200893621, −4.742619946685422, −4.057019028488575, −3.775406677686607, −2.924006124652111, −2.309216814637438, −1.494859673113324, −0.7867880579591988, 0,
0.7867880579591988, 1.494859673113324, 2.309216814637438, 2.924006124652111, 3.775406677686607, 4.057019028488575, 4.742619946685422, 5.358520200893621, 5.827847407965788, 6.326369505190371, 6.936746840689345, 7.473881381258904, 7.952145802025208, 8.383157733077394, 9.091144897038380, 9.611830513794850, 9.996972590913863, 10.75249877646945, 10.98710244665513, 11.55337585922759, 11.99439138492374, 12.39134954867641, 12.91818577976065, 13.52957409590408, 13.99057655542971
