| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s − 11-s + 2·13-s − 15-s − 2·17-s − 4·19-s − 4·21-s + 25-s − 27-s + 6·29-s + 8·31-s + 33-s + 4·35-s + 2·37-s − 2·39-s − 2·41-s − 4·43-s + 45-s + 9·49-s + 2·51-s + 6·53-s − 55-s + 4·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-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 + 1.11·29-s + 1.43·31-s + 0.174·33-s + 0.676·35-s + 0.328·37-s − 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.149·45-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 0.134·55-s + 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.191829112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.191829112\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.436014473790375803610234707099, −7.51078156006080628572337641472, −6.57372547437956407618343333440, −6.12104696706284593873842429125, −5.11308045142133625548253371673, −4.74752608600863894897799004131, −3.94602594161662769604748692192, −2.61402347693219480955223270217, −1.80574850351898704525593262124, −0.864722347053800555481702543043,
0.864722347053800555481702543043, 1.80574850351898704525593262124, 2.61402347693219480955223270217, 3.94602594161662769604748692192, 4.74752608600863894897799004131, 5.11308045142133625548253371673, 6.12104696706284593873842429125, 6.57372547437956407618343333440, 7.51078156006080628572337641472, 8.436014473790375803610234707099
