| L(s) = 1 | + 2·7-s + 11-s + 2·13-s + 2·17-s + 6·19-s + 4·23-s − 5·25-s + 10·29-s − 8·31-s + 6·37-s + 6·41-s − 6·43-s − 3·49-s − 12·53-s + 12·59-s + 6·61-s − 8·67-s + 8·71-s − 10·73-s + 2·77-s + 10·79-s + 4·83-s − 16·89-s + 4·91-s − 14·97-s + 2·101-s + 4·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s + 0.554·13-s + 0.485·17-s + 1.37·19-s + 0.834·23-s − 25-s + 1.85·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s − 0.914·43-s − 3/7·49-s − 1.64·53-s + 1.56·59-s + 0.768·61-s − 0.977·67-s + 0.949·71-s − 1.17·73-s + 0.227·77-s + 1.12·79-s + 0.439·83-s − 1.69·89-s + 0.419·91-s − 1.42·97-s + 0.199·101-s + 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.607117659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.607117659\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.036490883162075781681639493596, −7.41326118724701745885048975491, −6.66538518976282866675033180421, −5.82432976787725516205810348969, −5.19971191855077075608148012225, −4.46676966413404096333033613129, −3.58880564759217801399216043398, −2.84431585037210729336635244405, −1.67526754475183600725815923432, −0.911750413347333205165195473846,
0.911750413347333205165195473846, 1.67526754475183600725815923432, 2.84431585037210729336635244405, 3.58880564759217801399216043398, 4.46676966413404096333033613129, 5.19971191855077075608148012225, 5.82432976787725516205810348969, 6.66538518976282866675033180421, 7.41326118724701745885048975491, 8.036490883162075781681639493596
