| L(s) = 1 | − 3-s + 7-s + 9-s − 6·11-s − 2·13-s − 21-s − 6·23-s − 5·25-s − 27-s − 6·29-s − 8·31-s + 6·33-s − 2·37-s + 2·39-s − 12·41-s − 4·43-s + 12·47-s + 49-s + 6·53-s − 10·61-s + 63-s − 8·67-s + 6·69-s − 6·71-s − 10·73-s + 5·75-s − 6·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.218·21-s − 1.25·23-s − 25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.04·33-s − 0.328·37-s + 0.320·39-s − 1.87·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.28·61-s + 0.125·63-s − 0.977·67-s + 0.722·69-s − 0.712·71-s − 1.17·73-s + 0.577·75-s − 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−15.57519334460699, −15.23340259623677, −14.75248225522052, −13.91263269680853, −13.48486077154235, −13.07660703150546, −12.31701715709444, −12.03067718386861, −11.37529114107759, −10.79985623658582, −10.28704116149626, −9.984698028058618, −9.216345078405396, −8.500435077349755, −7.917272343501993, −7.379950768631112, −7.076679321581204, −5.913865753766269, −5.642266454946787, −5.154513544659710, −4.427070919548248, −3.788792829118432, −2.934628808897710, −2.114035956311888, −1.620993897777725, 0, 0,
1.620993897777725, 2.114035956311888, 2.934628808897710, 3.788792829118432, 4.427070919548248, 5.154513544659710, 5.642266454946787, 5.913865753766269, 7.076679321581204, 7.379950768631112, 7.917272343501993, 8.500435077349755, 9.216345078405396, 9.984698028058618, 10.28704116149626, 10.79985623658582, 11.37529114107759, 12.03067718386861, 12.31701715709444, 13.07660703150546, 13.48486077154235, 13.91263269680853, 14.75248225522052, 15.23340259623677, 15.57519334460699
