| L(s) = 1 | − 3-s + 9-s + 4·11-s − 2·13-s + 2·17-s + 4·19-s + 8·23-s − 27-s + 6·29-s − 8·31-s − 4·33-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s − 2·51-s + 2·53-s − 4·57-s − 4·59-s + 2·61-s + 4·67-s − 8·69-s + 8·71-s + 10·73-s − 8·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.280·51-s + 0.274·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.488·67-s − 0.963·69-s + 0.949·71-s + 1.17·73-s − 0.900·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.279550535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.279550535\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.15814000313491, −14.54015046337484, −14.22222427187212, −13.64140932005695, −12.79397814403659, −12.56895507421066, −11.84583449273832, −11.54301820761030, −10.92863880481933, −10.37098223423432, −9.721109928460394, −9.231400422182988, −8.791000323463798, −7.968061500225451, −7.199954245786336, −6.976165337503814, −6.286517914378789, −5.567340254086141, −5.047680771196692, −4.516152268909338, −3.598197994882597, −3.195291957952036, −2.161545243570202, −1.286357493252550, −0.6771319214532245,
0.6771319214532245, 1.286357493252550, 2.161545243570202, 3.195291957952036, 3.598197994882597, 4.516152268909338, 5.047680771196692, 5.567340254086141, 6.286517914378789, 6.976165337503814, 7.199954245786336, 7.968061500225451, 8.791000323463798, 9.231400422182988, 9.721109928460394, 10.37098223423432, 10.92863880481933, 11.54301820761030, 11.84583449273832, 12.56895507421066, 12.79397814403659, 13.64140932005695, 14.22222427187212, 14.54015046337484, 15.15814000313491
