| L(s) = 1 | + 7-s + 2·11-s − 6·17-s + 6·19-s + 8·23-s − 6·29-s + 6·31-s + 10·37-s − 2·41-s − 4·43-s − 8·47-s + 49-s − 12·53-s + 12·59-s − 10·61-s − 4·67-s − 2·71-s − 4·73-s + 2·77-s − 4·79-s − 12·83-s − 6·89-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.603·11-s − 1.45·17-s + 1.37·19-s + 1.66·23-s − 1.11·29-s + 1.07·31-s + 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 1.64·53-s + 1.56·59-s − 1.28·61-s − 0.488·67-s − 0.237·71-s − 0.468·73-s + 0.227·77-s − 0.450·79-s − 1.31·83-s − 0.635·89-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−14.67268498492094, −14.40710345702539, −13.64467836890464, −13.20034531382196, −12.94349745598557, −12.10334091675269, −11.52520619388865, −11.24908878072244, −10.88766618165646, −9.925626684580617, −9.611001939513318, −9.019114071665892, −8.584616119318802, −7.914918937532760, −7.341750395688189, −6.823445058429782, −6.310760229058009, −5.643218930480228, −4.889650220052386, −4.594089300742269, −3.843054182144941, −3.072057386535606, −2.598614994933068, −1.587312494207829, −1.116236206233067, 0,
1.116236206233067, 1.587312494207829, 2.598614994933068, 3.072057386535606, 3.843054182144941, 4.594089300742269, 4.889650220052386, 5.643218930480228, 6.310760229058009, 6.823445058429782, 7.341750395688189, 7.914918937532760, 8.584616119318802, 9.019114071665892, 9.611001939513318, 9.925626684580617, 10.88766618165646, 11.24908878072244, 11.52520619388865, 12.10334091675269, 12.94349745598557, 13.20034531382196, 13.64467836890464, 14.40710345702539, 14.67268498492094
