| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 11-s − 15-s + 19-s − 21-s + 4·23-s + 25-s − 27-s + 4·29-s + 2·31-s + 33-s + 35-s − 37-s − 6·41-s − 10·43-s + 45-s + 7·47-s − 6·49-s + 53-s − 55-s − 57-s − 4·59-s − 14·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.258·15-s + 0.229·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.359·31-s + 0.174·33-s + 0.169·35-s − 0.164·37-s − 0.937·41-s − 1.52·43-s + 0.149·45-s + 1.02·47-s − 6/7·49-s + 0.137·53-s − 0.134·55-s − 0.132·57-s − 0.520·59-s − 1.79·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.84074722089237, −15.46042437549782, −14.85227520784897, −14.28873421366320, −13.62049041680857, −13.25943599480712, −12.64501808325318, −11.91857609265703, −11.66122007170464, −10.88120845083801, −10.38176974108907, −10.00774116844818, −9.182343515894495, −8.704670493825034, −7.992269228310207, −7.354115815822594, −6.739209105359116, −6.146238723052795, −5.525476409562016, −4.836974243688281, −4.519411049519499, −3.388318191400184, −2.805253635783694, −1.793744602093880, −1.156468665109177, 0,
1.156468665109177, 1.793744602093880, 2.805253635783694, 3.388318191400184, 4.519411049519499, 4.836974243688281, 5.525476409562016, 6.146238723052795, 6.739209105359116, 7.354115815822594, 7.992269228310207, 8.704670493825034, 9.182343515894495, 10.00774116844818, 10.38176974108907, 10.88120845083801, 11.66122007170464, 11.91857609265703, 12.64501808325318, 13.25943599480712, 13.62049041680857, 14.28873421366320, 14.85227520784897, 15.46042437549782, 15.84074722089237
