| L(s) = 1 | + 2-s − 4-s − 5-s − 3·8-s − 10-s − 4·13-s − 16-s − 4·17-s − 4·19-s + 20-s − 8·23-s + 25-s − 4·26-s − 2·29-s + 2·31-s + 5·32-s − 4·34-s + 6·37-s − 4·38-s + 3·40-s + 6·41-s − 8·46-s − 12·47-s − 7·49-s + 50-s + 4·52-s + 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s − 1.10·13-s − 1/4·16-s − 0.970·17-s − 0.917·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.784·26-s − 0.371·29-s + 0.359·31-s + 0.883·32-s − 0.685·34-s + 0.986·37-s − 0.648·38-s + 0.474·40-s + 0.937·41-s − 1.17·46-s − 1.75·47-s − 49-s + 0.141·50-s + 0.554·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−12.84197402104237, −12.33706602796713, −11.88053907062726, −11.60192709794656, −11.03852085870482, −10.48899874410066, −9.959477330799626, −9.581910346654228, −9.154195794370586, −8.569765410582406, −8.194826835939965, −7.719377547877275, −7.257183676923274, −6.519937357337876, −6.192900335093404, −5.804140827237406, −5.063133433038671, −4.511974293496846, −4.459223325570348, −3.853516144216316, −3.273510860353006, −2.660731144303180, −2.208686258119073, −1.534412114434025, −0.4497203373100922, 0,
0.4497203373100922, 1.534412114434025, 2.208686258119073, 2.660731144303180, 3.273510860353006, 3.853516144216316, 4.459223325570348, 4.511974293496846, 5.063133433038671, 5.804140827237406, 6.192900335093404, 6.519937357337876, 7.257183676923274, 7.719377547877275, 8.194826835939965, 8.569765410582406, 9.154195794370586, 9.581910346654228, 9.959477330799626, 10.48899874410066, 11.03852085870482, 11.60192709794656, 11.88053907062726, 12.33706602796713, 12.84197402104237
