| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 15-s + 6·17-s + 2·19-s − 2·21-s + 25-s − 27-s + 2·29-s + 6·31-s − 2·35-s − 6·41-s − 4·43-s − 45-s − 12·47-s − 3·49-s − 6·51-s − 14·53-s − 2·57-s − 4·59-s − 6·61-s + 2·63-s + 2·67-s + 8·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.258·15-s + 1.45·17-s + 0.458·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.338·35-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.840·51-s − 1.92·53-s − 0.264·57-s − 0.520·59-s − 0.768·61-s + 0.251·63-s + 0.244·67-s + 0.949·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.82363767765882, −15.53326713849958, −14.80195688817467, −14.25076221853170, −13.94114383765435, −13.01962828360907, −12.60166329155052, −11.80504629579796, −11.71270144408514, −11.07192168586658, −10.39840915821479, −9.881379902138004, −9.377197976995821, −8.302361377854253, −8.104960711675817, −7.517330012625405, −6.725221856422907, −6.230764730530877, −5.360059737245222, −4.947796894655668, −4.377714626912684, −3.429412446488659, −2.949993080634110, −1.671990863236211, −1.152500044214598, 0,
1.152500044214598, 1.671990863236211, 2.949993080634110, 3.429412446488659, 4.377714626912684, 4.947796894655668, 5.360059737245222, 6.230764730530877, 6.725221856422907, 7.517330012625405, 8.104960711675817, 8.302361377854253, 9.377197976995821, 9.881379902138004, 10.39840915821479, 11.07192168586658, 11.71270144408514, 11.80504629579796, 12.60166329155052, 13.01962828360907, 13.94114383765435, 14.25076221853170, 14.80195688817467, 15.53326713849958, 15.82363767765882
