| L(s) = 1 | + 3-s + 3·5-s + 9-s + 3·11-s − 2·13-s + 3·15-s + 2·19-s + 4·23-s + 4·25-s + 27-s − 3·29-s − 3·31-s + 3·33-s − 6·37-s − 2·39-s − 4·41-s + 10·43-s + 3·45-s − 8·47-s − 5·53-s + 9·55-s + 2·57-s + 13·59-s − 6·65-s − 8·67-s + 4·69-s − 10·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1/3·9-s + 0.904·11-s − 0.554·13-s + 0.774·15-s + 0.458·19-s + 0.834·23-s + 4/5·25-s + 0.192·27-s − 0.557·29-s − 0.538·31-s + 0.522·33-s − 0.986·37-s − 0.320·39-s − 0.624·41-s + 1.52·43-s + 0.447·45-s − 1.16·47-s − 0.686·53-s + 1.21·55-s + 0.264·57-s + 1.69·59-s − 0.744·65-s − 0.977·67-s + 0.481·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 339864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 339864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.87074671844250, −12.40451957856036, −12.00919275040667, −11.40157987141663, −10.93380439326280, −10.44057612621626, −9.921634929009238, −9.542963550688387, −9.256561201495591, −8.824970286343757, −8.359590385056389, −7.664394150444019, −7.228140415750641, −6.733059177156759, −6.421675579204317, −5.609613299824795, −5.449992926065423, −4.845333505598560, −4.192188118115492, −3.716292003207429, −3.009577258795239, −2.684665434357500, −1.912938115063973, −1.593649108719969, −1.026172650148367, 0,
1.026172650148367, 1.593649108719969, 1.912938115063973, 2.684665434357500, 3.009577258795239, 3.716292003207429, 4.192188118115492, 4.845333505598560, 5.449992926065423, 5.609613299824795, 6.421675579204317, 6.733059177156759, 7.228140415750641, 7.664394150444019, 8.359590385056389, 8.824970286343757, 9.256561201495591, 9.542963550688387, 9.921634929009238, 10.44057612621626, 10.93380439326280, 11.40157987141663, 12.00919275040667, 12.40451957856036, 12.87074671844250
