| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s − 5·7-s + 3·8-s + 9-s − 6·11-s + 12-s + 5·14-s − 16-s − 18-s − 19-s + 5·21-s + 6·22-s − 5·23-s − 3·24-s − 27-s + 5·28-s + 8·29-s − 5·31-s − 5·32-s + 6·33-s − 36-s + 10·37-s + 38-s + 4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.88·7-s + 1.06·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 1.33·14-s − 1/4·16-s − 0.235·18-s − 0.229·19-s + 1.09·21-s + 1.27·22-s − 1.04·23-s − 0.612·24-s − 0.192·27-s + 0.944·28-s + 1.48·29-s − 0.898·31-s − 0.883·32-s + 1.04·33-s − 1/6·36-s + 1.64·37-s + 0.162·38-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.03819392030568, −12.72971164896976, −12.34567908490858, −11.78763318107042, −10.98653599739123, −10.59929148892378, −10.28803547224516, −9.881224960430397, −9.533363737354367, −9.023309884248406, −8.459660585543719, −7.978888719689162, −7.374131130846251, −7.213257156730512, −6.356908949037378, −5.862385338588369, −5.744909348790996, −4.796857034993013, −4.524910855756737, −3.840897323862669, −3.257838379017254, −2.606863832482673, −2.206190084124525, −1.106556052740987, −0.4784311294800558, 0,
0.4784311294800558, 1.106556052740987, 2.206190084124525, 2.606863832482673, 3.257838379017254, 3.840897323862669, 4.524910855756737, 4.796857034993013, 5.744909348790996, 5.862385338588369, 6.356908949037378, 7.213257156730512, 7.374131130846251, 7.978888719689162, 8.459660585543719, 9.023309884248406, 9.533363737354367, 9.881224960430397, 10.28803547224516, 10.59929148892378, 10.98653599739123, 11.78763318107042, 12.34567908490858, 12.72971164896976, 13.03819392030568
