| L(s) = 1 | + 3-s − 5-s + 7-s − 2·9-s + 5·11-s + 13-s − 15-s + 3·17-s + 6·19-s + 21-s + 6·23-s + 25-s − 5·27-s − 9·29-s + 5·33-s − 35-s + 6·37-s + 39-s + 8·41-s − 6·43-s + 2·45-s − 3·47-s + 49-s + 3·51-s − 12·53-s − 5·55-s + 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 1.50·11-s + 0.277·13-s − 0.258·15-s + 0.727·17-s + 1.37·19-s + 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 0.870·33-s − 0.169·35-s + 0.986·37-s + 0.160·39-s + 1.24·41-s − 0.914·43-s + 0.298·45-s − 0.437·47-s + 1/7·49-s + 0.420·51-s − 1.64·53-s − 0.674·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.792421847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.792421847\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−11.09811082697236396880583849134, −9.460974751463711551486664853134, −9.184452272376042915548869165944, −8.055738154784870307941541346096, −7.40499721789098877567948378074, −6.21937523368271298874577438547, −5.14802755062881381906448220617, −3.84388371346561797293344696886, −3.05770384156940941860061753705, −1.34060858351327769859608529280,
1.34060858351327769859608529280, 3.05770384156940941860061753705, 3.84388371346561797293344696886, 5.14802755062881381906448220617, 6.21937523368271298874577438547, 7.40499721789098877567948378074, 8.055738154784870307941541346096, 9.184452272376042915548869165944, 9.460974751463711551486664853134, 11.09811082697236396880583849134
