| L(s) = 1 | − 2·5-s + 2·17-s − 4·19-s + 23-s − 25-s + 10·29-s + 6·31-s + 2·37-s + 4·43-s + 6·47-s + 2·53-s + 2·59-s − 2·61-s − 8·71-s + 4·73-s − 4·79-s − 4·85-s − 6·89-s + 8·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·115-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.485·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s + 1.85·29-s + 1.07·31-s + 0.328·37-s + 0.609·43-s + 0.875·47-s + 0.274·53-s + 0.260·59-s − 0.256·61-s − 0.949·71-s + 0.468·73-s − 0.450·79-s − 0.433·85-s − 0.635·89-s + 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.186·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.53819739624580, −12.98198223092430, −12.41018573993430, −12.02734666489040, −11.79606057190416, −11.09788060709296, −10.67928779234693, −10.23002952565339, −9.756575623460383, −9.112394863035090, −8.605476347876071, −8.081537594620309, −7.913956142937953, −7.112016104843519, −6.789189704911103, −6.141489601260641, −5.700574809270147, −4.960911064489536, −4.389109439293550, −4.126026839056667, −3.434824027830025, −2.761062352652501, −2.392319025215966, −1.371023632255332, −0.8101683502741158, 0,
0.8101683502741158, 1.371023632255332, 2.392319025215966, 2.761062352652501, 3.434824027830025, 4.126026839056667, 4.389109439293550, 4.960911064489536, 5.700574809270147, 6.141489601260641, 6.789189704911103, 7.112016104843519, 7.913956142937953, 8.081537594620309, 8.605476347876071, 9.112394863035090, 9.756575623460383, 10.23002952565339, 10.67928779234693, 11.09788060709296, 11.79606057190416, 12.02734666489040, 12.41018573993430, 12.98198223092430, 13.53819739624580
