| L(s) = 1 | − 3-s + 9-s + 6·11-s + 2·13-s + 6·17-s − 2·19-s − 23-s − 5·25-s − 27-s + 6·29-s + 8·31-s − 6·33-s − 8·37-s − 2·39-s − 6·41-s + 2·43-s − 6·51-s + 12·53-s + 2·57-s + 8·61-s − 10·67-s + 69-s − 14·73-s + 5·75-s − 8·79-s + 81-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s − 0.208·23-s − 25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 1.04·33-s − 1.31·37-s − 0.320·39-s − 0.937·41-s + 0.304·43-s − 0.840·51-s + 1.64·53-s + 0.264·57-s + 1.02·61-s − 1.22·67-s + 0.120·69-s − 1.63·73-s + 0.577·75-s − 0.900·79-s + 1/9·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.26109959995514, −12.59983744890892, −12.01040587099704, −11.84028356386808, −11.68673455697390, −10.88843379335536, −10.32589561561572, −9.974187283454267, −9.696521819452137, −8.774976499948771, −8.650293680525611, −8.121199091215605, −7.310463141569925, −7.032732697923326, −6.374572350571588, −6.083675244344131, −5.613036045874690, −4.985255045376057, −4.310279486754562, −3.961262226538779, −3.447738873437196, −2.817886937321449, −1.942969225450907, −1.258658605245200, −1.026463233122999, 0,
1.026463233122999, 1.258658605245200, 1.942969225450907, 2.817886937321449, 3.447738873437196, 3.961262226538779, 4.310279486754562, 4.985255045376057, 5.613036045874690, 6.083675244344131, 6.374572350571588, 7.032732697923326, 7.310463141569925, 8.121199091215605, 8.650293680525611, 8.774976499948771, 9.696521819452137, 9.974187283454267, 10.32589561561572, 10.88843379335536, 11.68673455697390, 11.84028356386808, 12.01040587099704, 12.59983744890892, 13.26109959995514
