| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 6·11-s − 4·13-s − 15-s + 2·19-s + 21-s + 25-s − 27-s − 6·29-s + 10·31-s + 6·33-s − 35-s − 2·37-s + 4·39-s + 6·41-s − 4·43-s + 45-s + 49-s − 12·53-s − 6·55-s − 2·57-s − 14·61-s − 63-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s − 0.258·15-s + 0.458·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.79·31-s + 1.04·33-s − 0.169·35-s − 0.328·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s − 1.64·53-s − 0.809·55-s − 0.264·57-s − 1.79·61-s − 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.58732944988135, −13.33363235957278, −12.84104771291891, −12.27909789720312, −12.06817889367567, −11.30935248490078, −10.77511472449391, −10.47045355785602, −9.833009186119678, −9.660666779418086, −9.046450306597025, −8.258798109773194, −7.729000381258833, −7.466371994828787, −6.813065830921071, −6.162682296941546, −5.800919552275475, −5.136285303464024, −4.846357557248881, −4.321053935979463, −3.271508983983105, −2.899946629123645, −2.289559891131890, −1.647054009313215, −0.6439481210747235, 0,
0.6439481210747235, 1.647054009313215, 2.289559891131890, 2.899946629123645, 3.271508983983105, 4.321053935979463, 4.846357557248881, 5.136285303464024, 5.800919552275475, 6.162682296941546, 6.813065830921071, 7.466371994828787, 7.729000381258833, 8.258798109773194, 9.046450306597025, 9.660666779418086, 9.833009186119678, 10.47045355785602, 10.77511472449391, 11.30935248490078, 12.06817889367567, 12.27909789720312, 12.84104771291891, 13.33363235957278, 13.58732944988135
