| L(s) = 1 | + 4·5-s − 7-s + 4·11-s − 13-s + 3·17-s + 2·19-s − 3·23-s + 11·25-s − 9·29-s − 31-s − 4·35-s + 10·37-s + 10·41-s − 43-s − 10·47-s + 49-s + 9·53-s + 16·55-s − 3·59-s − 6·61-s − 4·65-s − 13·67-s − 13·71-s + 10·73-s − 4·77-s + 10·79-s + 4·83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s + 1.20·11-s − 0.277·13-s + 0.727·17-s + 0.458·19-s − 0.625·23-s + 11/5·25-s − 1.67·29-s − 0.179·31-s − 0.676·35-s + 1.64·37-s + 1.56·41-s − 0.152·43-s − 1.45·47-s + 1/7·49-s + 1.23·53-s + 2.15·55-s − 0.390·59-s − 0.768·61-s − 0.496·65-s − 1.58·67-s − 1.54·71-s + 1.17·73-s − 0.455·77-s + 1.12·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.463254201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.463254201\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−9.442320445686517988686722905025, −9.124477874703552024915151089158, −7.81195631666782754127091281103, −6.89975695051450867215286186940, −5.97765521850791274570773320089, −5.71783228302031072107350458669, −4.48329735587609522917856923409, −3.32204369752315772488428657976, −2.21234938564256767010135396120, −1.25282736319694142732812877021,
1.25282736319694142732812877021, 2.21234938564256767010135396120, 3.32204369752315772488428657976, 4.48329735587609522917856923409, 5.71783228302031072107350458669, 5.97765521850791274570773320089, 6.89975695051450867215286186940, 7.81195631666782754127091281103, 9.124477874703552024915151089158, 9.442320445686517988686722905025
