| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 2·13-s + 15-s + 2·17-s + 4·19-s − 4·21-s + 25-s + 27-s − 2·29-s − 4·35-s − 10·37-s + 2·39-s − 6·41-s − 43-s + 45-s + 8·47-s + 9·49-s + 2·51-s − 2·53-s + 4·57-s − 2·61-s − 4·63-s + 2·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.676·35-s − 1.64·37-s + 0.320·39-s − 0.937·41-s − 0.152·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.274·53-s + 0.529·57-s − 0.256·61-s − 0.503·63-s + 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.510568387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.510568387\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.71096224393060, −14.07954280948729, −13.61872005268135, −13.37200988149634, −12.77318486089200, −12.16047887353588, −11.88899014620355, −10.88677981750141, −10.43042635780376, −9.941056829041787, −9.447010771301151, −9.016848015168065, −8.531426047782179, −7.701943335854391, −7.250070838651722, −6.579166095442009, −6.190181151789531, −5.472233393623442, −4.963938649509396, −3.916124163048776, −3.433301670503265, −3.072927841627668, −2.242289338784338, −1.470560924167544, −0.5585974240581658,
0.5585974240581658, 1.470560924167544, 2.242289338784338, 3.072927841627668, 3.433301670503265, 3.916124163048776, 4.963938649509396, 5.472233393623442, 6.190181151789531, 6.579166095442009, 7.250070838651722, 7.701943335854391, 8.531426047782179, 9.016848015168065, 9.447010771301151, 9.941056829041787, 10.43042635780376, 10.88677981750141, 11.88899014620355, 12.16047887353588, 12.77318486089200, 13.37200988149634, 13.61872005268135, 14.07954280948729, 14.71096224393060
