| L(s) = 1 | − 3-s − 5-s + 9-s − 2·13-s + 15-s − 6·17-s − 4·19-s + 4·23-s + 25-s − 27-s − 6·29-s − 6·37-s + 2·39-s − 6·41-s + 4·43-s − 45-s − 8·47-s + 6·51-s − 14·53-s + 4·57-s − 4·59-s − 2·61-s + 2·65-s − 12·67-s − 4·69-s − 12·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 0.840·51-s − 1.92·53-s + 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.481·69-s − 1.42·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−15.04758958910235, −14.78741248316758, −14.13462984385303, −13.31555059161500, −13.06769166755566, −12.57521963496391, −11.92564387635693, −11.51165997264007, −10.93216080339024, −10.64142951950470, −10.00569089090765, −9.247462072138405, −8.892206735863323, −8.314646801015116, −7.503818181362170, −7.203988867803930, −6.469445567146594, −6.157493949483426, −5.270486409017460, −4.704069373556620, −4.382670736603775, −3.544687324621981, −2.898219013296051, −2.019098627016955, −1.434815263553232, 0, 0,
1.434815263553232, 2.019098627016955, 2.898219013296051, 3.544687324621981, 4.382670736603775, 4.704069373556620, 5.270486409017460, 6.157493949483426, 6.469445567146594, 7.203988867803930, 7.503818181362170, 8.314646801015116, 8.892206735863323, 9.247462072138405, 10.00569089090765, 10.64142951950470, 10.93216080339024, 11.51165997264007, 11.92564387635693, 12.57521963496391, 13.06769166755566, 13.31555059161500, 14.13462984385303, 14.78741248316758, 15.04758958910235
