| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s + 13-s − 15-s + 16-s + 3·17-s − 18-s − 7·19-s − 20-s − 24-s + 25-s − 26-s + 27-s − 6·29-s + 30-s + 2·31-s − 32-s − 3·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.60·19-s − 0.223·20-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s + 0.359·31-s − 0.176·32-s − 0.514·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−16.05213398734789, −15.42550493256142, −14.92506069609432, −14.58362459224894, −13.90535967931033, −13.10007742033900, −12.77807484531562, −12.11652685244610, −11.43212356784389, −11.00448991925595, −10.24987682260963, −9.947989628340742, −9.068055277601590, −8.701781371100542, −8.173791931040209, −7.537141647140021, −7.119606268684237, −6.275614681859268, −5.783551727334502, −4.788031890944787, −4.058243069505522, −3.488242297460384, −2.651466976866246, −1.967267152521650, −1.081752531390253, 0,
1.081752531390253, 1.967267152521650, 2.651466976866246, 3.488242297460384, 4.058243069505522, 4.788031890944787, 5.783551727334502, 6.275614681859268, 7.119606268684237, 7.537141647140021, 8.173791931040209, 8.701781371100542, 9.068055277601590, 9.947989628340742, 10.24987682260963, 11.00448991925595, 11.43212356784389, 12.11652685244610, 12.77807484531562, 13.10007742033900, 13.90535967931033, 14.58362459224894, 14.92506069609432, 15.42550493256142, 16.05213398734789
