| L(s) = 1 | + 3-s + 3·5-s + 9-s − 3·11-s + 13-s + 3·15-s + 3·17-s − 7·19-s + 9·23-s + 4·25-s + 27-s + 9·29-s + 4·31-s − 3·33-s + 7·37-s + 39-s − 12·41-s + 43-s + 3·45-s + 3·51-s + 6·53-s − 9·55-s − 7·57-s + 12·59-s − 61-s + 3·65-s − 14·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.774·15-s + 0.727·17-s − 1.60·19-s + 1.87·23-s + 4/5·25-s + 0.192·27-s + 1.67·29-s + 0.718·31-s − 0.522·33-s + 1.15·37-s + 0.160·39-s − 1.87·41-s + 0.152·43-s + 0.447·45-s + 0.420·51-s + 0.824·53-s − 1.21·55-s − 0.927·57-s + 1.56·59-s − 0.128·61-s + 0.372·65-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.119218273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.119218273\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.34056040591426, −13.15584460516678, −12.95175399316984, −12.16372856308923, −11.69652819528175, −10.87321891774677, −10.53734046824544, −10.04293144070636, −9.860375508052879, −9.004014689845888, −8.718157536426655, −8.264697937555064, −7.717621993991108, −6.956998251868602, −6.564111114112639, −6.103317260499261, −5.440223337996829, −4.952555315698154, −4.516697743523945, −3.686603750004336, −2.938040263795360, −2.611573294456065, −2.058117703078089, −1.306899984902259, −0.6814301971819957,
0.6814301971819957, 1.306899984902259, 2.058117703078089, 2.611573294456065, 2.938040263795360, 3.686603750004336, 4.516697743523945, 4.952555315698154, 5.440223337996829, 6.103317260499261, 6.564111114112639, 6.956998251868602, 7.717621993991108, 8.264697937555064, 8.718157536426655, 9.004014689845888, 9.860375508052879, 10.04293144070636, 10.53734046824544, 10.87321891774677, 11.69652819528175, 12.16372856308923, 12.95175399316984, 13.15584460516678, 13.34056040591426
