| L(s) = 1 | + 11-s − 2·13-s − 6·17-s − 4·19-s + 6·29-s + 8·31-s + 6·37-s − 10·41-s + 4·43-s + 8·47-s − 7·49-s + 10·53-s + 12·59-s − 6·61-s + 4·67-s + 14·73-s + 4·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.11·29-s + 1.43·31-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s + 1.37·53-s + 1.56·59-s − 0.768·61-s + 0.488·67-s + 1.63·73-s + 0.439·83-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.159129544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159129544\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.32266188235396, −12.86251234092003, −12.25198807906235, −11.92957276491796, −11.39586311205524, −10.94632064257912, −10.33248901077317, −10.07330930709665, −9.429364551660413, −8.911645393322992, −8.462201256926792, −8.108918036236459, −7.413290571374712, −6.762584180060723, −6.539712969631423, −6.056706422226451, −5.260810744724280, −4.729004453306235, −4.343041398232049, −3.817679380200039, −3.041175940501600, −2.299048531755442, −2.177577691821275, −1.078443829543100, −0.4726096148470098,
0.4726096148470098, 1.078443829543100, 2.177577691821275, 2.299048531755442, 3.041175940501600, 3.817679380200039, 4.343041398232049, 4.729004453306235, 5.260810744724280, 6.056706422226451, 6.539712969631423, 6.762584180060723, 7.413290571374712, 8.108918036236459, 8.462201256926792, 8.911645393322992, 9.429364551660413, 10.07330930709665, 10.33248901077317, 10.94632064257912, 11.39586311205524, 11.92957276491796, 12.25198807906235, 12.86251234092003, 13.32266188235396
