| L(s) = 1 | + 3-s − 7-s + 9-s − 4·11-s − 6·17-s + 8·19-s − 21-s + 4·23-s − 5·25-s + 27-s + 2·29-s − 4·33-s − 8·37-s − 8·43-s + 4·47-s + 49-s − 6·51-s − 10·53-s + 8·57-s + 4·59-s + 6·61-s − 63-s + 8·67-s + 4·69-s − 4·71-s + 4·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.45·17-s + 1.83·19-s − 0.218·21-s + 0.834·23-s − 25-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 1.31·37-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.840·51-s − 1.37·53-s + 1.05·57-s + 0.520·59-s + 0.768·61-s − 0.125·63-s + 0.977·67-s + 0.481·69-s − 0.474·71-s + 0.468·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.917002049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917002049\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−15.92595174876078, −15.59600136883362, −15.23748037845246, −14.36148615528719, −13.80212629889515, −13.28252465115729, −13.08670728455094, −12.15467508518399, −11.70213797823368, −10.88299941486669, −10.47544656429726, −9.643463203217358, −9.378655343407923, −8.563941319871449, −8.046155678154763, −7.379901782848071, −6.869792961015851, −6.144870613466606, −5.105980847418114, −4.985342984699040, −3.802273092230758, −3.248001046509259, −2.532382120215165, −1.806791305034471, −0.5716684349568890,
0.5716684349568890, 1.806791305034471, 2.532382120215165, 3.248001046509259, 3.802273092230758, 4.985342984699040, 5.105980847418114, 6.144870613466606, 6.869792961015851, 7.379901782848071, 8.046155678154763, 8.563941319871449, 9.378655343407923, 9.643463203217358, 10.47544656429726, 10.88299941486669, 11.70213797823368, 12.15467508518399, 13.08670728455094, 13.28252465115729, 13.80212629889515, 14.36148615528719, 15.23748037845246, 15.59600136883362, 15.92595174876078
