| L(s) = 1 | − 3·3-s + 3·5-s + 7-s + 6·9-s + 4·13-s − 9·15-s − 3·17-s + 4·19-s − 3·21-s + 4·23-s + 4·25-s − 9·27-s + 3·29-s + 5·31-s + 3·35-s − 10·37-s − 12·39-s − 5·43-s + 18·45-s − 6·47-s + 49-s + 9·51-s + 3·53-s − 12·57-s + 6·59-s − 3·61-s + 6·63-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 0.377·7-s + 2·9-s + 1.10·13-s − 2.32·15-s − 0.727·17-s + 0.917·19-s − 0.654·21-s + 0.834·23-s + 4/5·25-s − 1.73·27-s + 0.557·29-s + 0.898·31-s + 0.507·35-s − 1.64·37-s − 1.92·39-s − 0.762·43-s + 2.68·45-s − 0.875·47-s + 1/7·49-s + 1.26·51-s + 0.412·53-s − 1.58·57-s + 0.781·59-s − 0.384·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084951757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084951757\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−13.67163909374911, −13.21596480728899, −13.00009145225136, −12.15075735245395, −11.83641800819399, −11.31639762507645, −10.91704373541281, −10.35851441764913, −10.11190073854576, −9.511592835197540, −8.871136937917399, −8.451922251526735, −7.627848683842845, −6.882620671511134, −6.539128960105439, −6.235334151609977, −5.469314362922917, −5.262906962619443, −4.800459026650839, −4.083438751767809, −3.320784812683206, −2.495922614102425, −1.611576691357941, −1.306236929138777, −0.5508661707446547,
0.5508661707446547, 1.306236929138777, 1.611576691357941, 2.495922614102425, 3.320784812683206, 4.083438751767809, 4.800459026650839, 5.262906962619443, 5.469314362922917, 6.235334151609977, 6.539128960105439, 6.882620671511134, 7.627848683842845, 8.451922251526735, 8.871136937917399, 9.511592835197540, 10.11190073854576, 10.35851441764913, 10.91704373541281, 11.31639762507645, 11.83641800819399, 12.15075735245395, 13.00009145225136, 13.21596480728899, 13.67163909374911
