| L(s) = 1 | + 3-s + 7-s + 9-s − 13-s − 3·19-s + 21-s + 4·23-s + 27-s + 4·29-s + 7·31-s + 6·37-s − 39-s + 6·41-s + 9·43-s + 6·47-s − 6·49-s − 2·53-s − 3·57-s − 10·59-s − 61-s + 63-s − 3·67-s + 4·69-s + 14·71-s − 10·73-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.277·13-s − 0.688·19-s + 0.218·21-s + 0.834·23-s + 0.192·27-s + 0.742·29-s + 1.25·31-s + 0.986·37-s − 0.160·39-s + 0.937·41-s + 1.37·43-s + 0.875·47-s − 6/7·49-s − 0.274·53-s − 0.397·57-s − 1.30·59-s − 0.128·61-s + 0.125·63-s − 0.366·67-s + 0.481·69-s + 1.66·71-s − 1.17·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.390498685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.390498685\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−8.986155872486398060955603454763, −8.138501938134350597335270637884, −7.61828348489898808289572591891, −6.68972285169098883459856444849, −5.92670201782955621143741934900, −4.77570500564148489314113812399, −4.24934920015302837146107605263, −3.04181470663783301255187706870, −2.29786134105577408045110307178, −1.00597629118782045123347487856,
1.00597629118782045123347487856, 2.29786134105577408045110307178, 3.04181470663783301255187706870, 4.24934920015302837146107605263, 4.77570500564148489314113812399, 5.92670201782955621143741934900, 6.68972285169098883459856444849, 7.61828348489898808289572591891, 8.138501938134350597335270637884, 8.986155872486398060955603454763
