| L(s) = 1 | − 2·2-s + 2·4-s + 4·11-s − 4·16-s + 17-s + 6·19-s − 8·22-s − 7·23-s − 5·25-s − 29-s − 9·31-s + 8·32-s − 2·34-s + 6·37-s − 12·38-s + 6·41-s − 9·43-s + 8·44-s + 14·46-s − 3·47-s + 10·50-s + 2·53-s + 2·58-s + 3·59-s − 3·61-s + 18·62-s − 8·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.20·11-s − 16-s + 0.242·17-s + 1.37·19-s − 1.70·22-s − 1.45·23-s − 25-s − 0.185·29-s − 1.61·31-s + 1.41·32-s − 0.342·34-s + 0.986·37-s − 1.94·38-s + 0.937·41-s − 1.37·43-s + 1.20·44-s + 2.06·46-s − 0.437·47-s + 1.41·50-s + 0.274·53-s + 0.262·58-s + 0.390·59-s − 0.384·61-s + 2.28·62-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22491 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22491 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−15.95611590635813, −15.50798086833228, −14.66454600404421, −14.21645679416450, −13.72617332687117, −13.09652921762561, −12.25393995669408, −11.81499144010261, −11.24292571410050, −10.88915633661815, −9.808636755020988, −9.720436420205926, −9.377497464589858, −8.445609974426517, −8.162322879605533, −7.372813468826135, −7.100235189561660, −6.208119951992608, −5.710227631756171, −4.836264429027323, −3.941994338436627, −3.534981057004461, −2.343135390370603, −1.680975638772707, −1.004477975379811, 0,
1.004477975379811, 1.680975638772707, 2.343135390370603, 3.534981057004461, 3.941994338436627, 4.836264429027323, 5.710227631756171, 6.208119951992608, 7.100235189561660, 7.372813468826135, 8.162322879605533, 8.445609974426517, 9.377497464589858, 9.720436420205926, 9.808636755020988, 10.88915633661815, 11.24292571410050, 11.81499144010261, 12.25393995669408, 13.09652921762561, 13.72617332687117, 14.21645679416450, 14.66454600404421, 15.50798086833228, 15.95611590635813
