| L(s) = 1 | + 3-s − 2·7-s + 9-s − 11-s + 13-s − 5·17-s + 4·19-s − 2·21-s + 27-s + 29-s + 5·31-s − 33-s + 37-s + 39-s + 10·41-s − 5·43-s − 3·47-s − 3·49-s − 5·51-s − 9·53-s + 4·57-s + 3·59-s − 13·61-s − 2·63-s − 5·67-s − 8·71-s − 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 1.21·17-s + 0.917·19-s − 0.436·21-s + 0.192·27-s + 0.185·29-s + 0.898·31-s − 0.174·33-s + 0.164·37-s + 0.160·39-s + 1.56·41-s − 0.762·43-s − 0.437·47-s − 3/7·49-s − 0.700·51-s − 1.23·53-s + 0.529·57-s + 0.390·59-s − 1.66·61-s − 0.251·63-s − 0.610·67-s − 0.949·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.83308381123784, −12.29968615883472, −11.81540859458350, −11.30019779281228, −10.94772742466138, −10.22626119983480, −10.08583305145403, −9.441449355390068, −8.996660025956765, −8.843301320786787, −8.020088433036456, −7.746823322931568, −7.290762240115944, −6.625010136118570, −6.268358710805205, −5.949219918665157, −5.081848198241622, −4.692140007958756, −4.239806452554373, −3.546461629462231, −3.094824434806876, −2.737130238522862, −2.079461126949906, −1.470390519268793, −0.7404532725041789, 0,
0.7404532725041789, 1.470390519268793, 2.079461126949906, 2.737130238522862, 3.094824434806876, 3.546461629462231, 4.239806452554373, 4.692140007958756, 5.081848198241622, 5.949219918665157, 6.268358710805205, 6.625010136118570, 7.290762240115944, 7.746823322931568, 8.020088433036456, 8.843301320786787, 8.996660025956765, 9.441449355390068, 10.08583305145403, 10.22626119983480, 10.94772742466138, 11.30019779281228, 11.81540859458350, 12.29968615883472, 12.83308381123784
