| L(s) = 1 | − 3-s + 5-s + 3·7-s + 9-s + 4·11-s − 5·13-s − 15-s + 7·19-s − 3·21-s − 6·23-s + 25-s − 27-s − 6·29-s − 5·31-s − 4·33-s + 3·35-s + 7·37-s + 5·39-s + 8·41-s + 11·43-s + 45-s − 6·47-s + 2·49-s − 6·53-s + 4·55-s − 7·57-s − 14·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.20·11-s − 1.38·13-s − 0.258·15-s + 1.60·19-s − 0.654·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.898·31-s − 0.696·33-s + 0.507·35-s + 1.15·37-s + 0.800·39-s + 1.24·41-s + 1.67·43-s + 0.149·45-s − 0.875·47-s + 2/7·49-s − 0.824·53-s + 0.539·55-s − 0.927·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−15.07745425192692, −14.58413247730438, −14.15154061887045, −13.96332850967949, −13.00725315274806, −12.45032908748969, −12.03624314198063, −11.47566661931975, −11.16663483095367, −10.52919550793469, −9.727718263660078, −9.412832484438179, −9.060996234772074, −7.949169731956558, −7.519139771390316, −7.296269545283043, −6.210033918863439, −5.902599633910518, −5.265827940652672, −4.591892313276582, −4.238789937736383, −3.315686454671495, −2.432573384922065, −1.660457305164539, −1.192249028405832, 0,
1.192249028405832, 1.660457305164539, 2.432573384922065, 3.315686454671495, 4.238789937736383, 4.591892313276582, 5.265827940652672, 5.902599633910518, 6.210033918863439, 7.296269545283043, 7.519139771390316, 7.949169731956558, 9.060996234772074, 9.412832484438179, 9.727718263660078, 10.52919550793469, 11.16663483095367, 11.47566661931975, 12.03624314198063, 12.45032908748969, 13.00725315274806, 13.96332850967949, 14.15154061887045, 14.58413247730438, 15.07745425192692
