| L(s) = 1 | − 3-s + 7-s + 9-s − 3·11-s − 4·13-s + 6·17-s − 4·19-s − 21-s + 3·23-s − 27-s − 3·29-s + 10·31-s + 3·33-s − 7·37-s + 4·39-s + 43-s − 12·47-s + 49-s − 6·51-s + 6·53-s + 4·57-s + 12·59-s + 4·61-s + 63-s + 7·67-s − 3·69-s − 9·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s + 0.625·23-s − 0.192·27-s − 0.557·29-s + 1.79·31-s + 0.522·33-s − 1.15·37-s + 0.640·39-s + 0.152·43-s − 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s + 1.56·59-s + 0.512·61-s + 0.125·63-s + 0.855·67-s − 0.361·69-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.15969332386046, −14.77286886174465, −14.35822955016360, −13.61629005904564, −13.02780105247871, −12.65133591696179, −11.98978036047711, −11.69438425331913, −11.02076223564246, −10.35181763228597, −10.06112934823578, −9.618102649482007, −8.614118632666447, −8.241858282117215, −7.610005388236954, −7.074460844725291, −6.538852533877368, −5.661298310741993, −5.291351886309724, −4.777021183885752, −4.130949536950722, −3.221203578058590, −2.597132503005154, −1.823975472909786, −0.9108536557755331, 0,
0.9108536557755331, 1.823975472909786, 2.597132503005154, 3.221203578058590, 4.130949536950722, 4.777021183885752, 5.291351886309724, 5.661298310741993, 6.538852533877368, 7.074460844725291, 7.610005388236954, 8.241858282117215, 8.614118632666447, 9.618102649482007, 10.06112934823578, 10.35181763228597, 11.02076223564246, 11.69438425331913, 11.98978036047711, 12.65133591696179, 13.02780105247871, 13.61629005904564, 14.35822955016360, 14.77286886174465, 15.15969332386046
