| L(s) = 1 | − 3-s − 2·4-s + 7-s − 2·9-s + 3·11-s + 2·12-s + 4·13-s + 4·16-s − 6·17-s + 2·19-s − 21-s − 6·23-s + 5·27-s − 2·28-s − 6·29-s − 4·31-s − 3·33-s + 4·36-s − 37-s − 4·39-s − 9·41-s − 8·43-s − 6·44-s − 3·47-s − 4·48-s − 6·49-s + 6·51-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.577·12-s + 1.10·13-s + 16-s − 1.45·17-s + 0.458·19-s − 0.218·21-s − 1.25·23-s + 0.962·27-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.522·33-s + 2/3·36-s − 0.164·37-s − 0.640·39-s − 1.40·41-s − 1.21·43-s − 0.904·44-s − 0.437·47-s − 0.577·48-s − 6/7·49-s + 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−9.591268537738103586397612813688, −8.647494221806725911176641352691, −8.390237773813478013059373004459, −6.96328856044475469810412834337, −6.03232690048232103384500782557, −5.31354420552212102801103714157, −4.28699193481010687037956905954, −3.49397830742570960606550504227, −1.63648598905395091233402759288, 0,
1.63648598905395091233402759288, 3.49397830742570960606550504227, 4.28699193481010687037956905954, 5.31354420552212102801103714157, 6.03232690048232103384500782557, 6.96328856044475469810412834337, 8.390237773813478013059373004459, 8.647494221806725911176641352691, 9.591268537738103586397612813688
