| L(s) = 1 | − 3-s + 9-s + 6·11-s + 2·13-s − 4·19-s − 6·23-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 + 6·53-s + 4·57-s + 10·61-s + 8·67-s + 6·69-s − 6·71-s − 10·73-s + 4·79-s + 81-s + 12·83-s − 6·87-s − 12·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.80·11-s + 0.554·13-s − 0.917·19-s − 1.25·23-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 + 0.824·53-s + 0.529·57-s + 1.28·61-s + 0.977·67-s + 0.722·69-s − 0.712·71-s − 1.17·73-s + 0.450·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 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.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.m |
| 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.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 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.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.63409763471826, −14.00108859728393, −13.64590361435583, −13.10759994685088, −12.33580849103462, −12.02548610623521, −11.56729426613310, −11.25368520380226, −10.35649564422114, −10.06095692007352, −9.633835864183000, −8.697934655408510, −8.512210458143715, −7.988287179343081, −6.914938880689256, −6.661382442008727, −6.289054374157980, −5.712198073393543, −4.847744289901779, −4.431494259642328, −3.780666108284599, −3.335964479388109, −2.278706994282686, −1.582587262734249, −1.008387036642399, 0,
1.008387036642399, 1.582587262734249, 2.278706994282686, 3.335964479388109, 3.780666108284599, 4.431494259642328, 4.847744289901779, 5.712198073393543, 6.289054374157980, 6.661382442008727, 6.914938880689256, 7.988287179343081, 8.512210458143715, 8.697934655408510, 9.633835864183000, 10.06095692007352, 10.35649564422114, 11.25368520380226, 11.56729426613310, 12.02548610623521, 12.33580849103462, 13.10759994685088, 13.64590361435583, 14.00108859728393, 14.63409763471826
