| L(s) = 1 | − 2·3-s − 3·5-s + 9-s − 6·11-s + 6·15-s + 5·17-s − 2·19-s − 8·23-s + 4·25-s + 4·27-s + 4·29-s + 3·31-s + 12·33-s − 37-s + 6·43-s − 3·45-s − 8·47-s − 10·51-s + 53-s + 18·55-s + 4·57-s + 12·59-s + 61-s − 7·67-s + 16·69-s − 8·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 1/3·9-s − 1.80·11-s + 1.54·15-s + 1.21·17-s − 0.458·19-s − 1.66·23-s + 4/5·25-s + 0.769·27-s + 0.742·29-s + 0.538·31-s + 2.08·33-s − 0.164·37-s + 0.914·43-s − 0.447·45-s − 1.16·47-s − 1.40·51-s + 0.137·53-s + 2.42·55-s + 0.529·57-s + 1.56·59-s + 0.128·61-s − 0.855·67-s + 1.92·69-s − 0.949·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7106492175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7106492175\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 229 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−12.91451573369406, −12.59095771769007, −12.00236267141282, −11.89919895489871, −11.34370838892705, −10.86114476699949, −10.34079650781623, −10.16278419966021, −9.578150647696745, −8.540083162134690, −8.268586166223245, −7.917273761410798, −7.434474311069760, −6.920474680274067, −6.237490959646675, −5.712819750480237, −5.441531577633858, −4.665640477017693, −4.474923465254447, −3.690377929915698, −3.144290304247787, −2.574555186642774, −1.805851667612700, −0.6976149633839411, −0.4069228870173896,
0.4069228870173896, 0.6976149633839411, 1.805851667612700, 2.574555186642774, 3.144290304247787, 3.690377929915698, 4.474923465254447, 4.665640477017693, 5.441531577633858, 5.712819750480237, 6.237490959646675, 6.920474680274067, 7.434474311069760, 7.917273761410798, 8.268586166223245, 8.540083162134690, 9.578150647696745, 10.16278419966021, 10.34079650781623, 10.86114476699949, 11.34370838892705, 11.89919895489871, 12.00236267141282, 12.59095771769007, 12.91451573369406
