| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 12-s − 6·13-s + 14-s + 16-s − 2·17-s − 18-s − 21-s − 2·23-s − 24-s − 5·25-s + 6·26-s + 27-s − 28-s + 29-s − 4·31-s − 32-s + 2·34-s + 36-s − 10·37-s − 6·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.218·21-s − 0.417·23-s − 0.204·24-s − 25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.185·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s − 1.64·37-s − 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 29 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−9.354760099557485020472356695239, −8.624822109963528950754972381040, −7.63275098194519557618093287447, −7.18709732341096879004434117262, −6.16807041787662090287830806021, −5.05137552310992527051824303456, −3.91140692867254512002888327424, −2.76191186124007964195873510012, −1.89167599998998245295952052999, 0,
1.89167599998998245295952052999, 2.76191186124007964195873510012, 3.91140692867254512002888327424, 5.05137552310992527051824303456, 6.16807041787662090287830806021, 7.18709732341096879004434117262, 7.63275098194519557618093287447, 8.624822109963528950754972381040, 9.354760099557485020472356695239
