| L(s) = 1 | − 2·3-s + 4·5-s + 9-s + 11-s + 2·13-s − 8·15-s − 2·17-s − 8·19-s + 11·25-s + 4·27-s + 6·29-s + 2·31-s − 2·33-s + 10·37-s − 4·39-s − 10·41-s − 8·43-s + 4·45-s + 6·47-s + 4·51-s + 6·53-s + 4·55-s + 16·57-s + 6·59-s − 2·61-s + 8·65-s − 12·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 2.06·15-s − 0.485·17-s − 1.83·19-s + 11/5·25-s + 0.769·27-s + 1.11·29-s + 0.359·31-s − 0.348·33-s + 1.64·37-s − 0.640·39-s − 1.56·41-s − 1.21·43-s + 0.596·45-s + 0.875·47-s + 0.560·51-s + 0.824·53-s + 0.539·55-s + 2.11·57-s + 0.781·59-s − 0.256·61-s + 0.992·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.121835576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.121835576\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.93238121041905, −14.44606656229042, −13.64123303230297, −13.49275782042169, −12.91710332673917, −12.32166827826775, −11.81742032817682, −11.13897845977879, −10.66812326002357, −10.26879771011957, −9.846426027928582, −9.009421659159333, −8.687550287415561, −8.071316776516373, −6.820148786845126, −6.597989903683323, −6.118604489145280, −5.779698588072384, −4.927246045518115, −4.666647377390459, −3.726964959508371, −2.675315414680075, −2.167268206916262, −1.387291600332863, −0.6066908492704788,
0.6066908492704788, 1.387291600332863, 2.167268206916262, 2.675315414680075, 3.726964959508371, 4.666647377390459, 4.927246045518115, 5.779698588072384, 6.118604489145280, 6.597989903683323, 6.820148786845126, 8.071316776516373, 8.687550287415561, 9.009421659159333, 9.846426027928582, 10.26879771011957, 10.66812326002357, 11.13897845977879, 11.81742032817682, 12.32166827826775, 12.91710332673917, 13.49275782042169, 13.64123303230297, 14.44606656229042, 14.93238121041905
