| L(s) = 1 | + 3·5-s + 3·7-s − 3·11-s − 6·13-s − 6·17-s + 2·19-s − 6·23-s + 4·25-s + 6·29-s − 3·31-s + 9·35-s − 6·37-s + 6·41-s + 8·43-s − 12·47-s + 2·49-s − 3·53-s − 9·55-s − 12·59-s − 18·65-s + 4·67-s + 6·71-s − 11·73-s − 9·77-s + 12·79-s + 9·83-s − 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.13·7-s − 0.904·11-s − 1.66·13-s − 1.45·17-s + 0.458·19-s − 1.25·23-s + 4/5·25-s + 1.11·29-s − 0.538·31-s + 1.52·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 2/7·49-s − 0.412·53-s − 1.21·55-s − 1.56·59-s − 2.23·65-s + 0.488·67-s + 0.712·71-s − 1.28·73-s − 1.02·77-s + 1.35·79-s + 0.987·83-s − 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−7.76620950749502010641913111207, −6.86121542489928951259646204249, −6.18462384653692041995716887886, −5.28332530727644253227148277492, −4.96986692409176893974637541358, −4.24443618730092525814161249234, −2.76963613538839295630327232914, −2.23959993180779682800139406382, −1.60447145814693122880088257777, 0,
1.60447145814693122880088257777, 2.23959993180779682800139406382, 2.76963613538839295630327232914, 4.24443618730092525814161249234, 4.96986692409176893974637541358, 5.28332530727644253227148277492, 6.18462384653692041995716887886, 6.86121542489928951259646204249, 7.76620950749502010641913111207
