| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s − 11-s + 3·13-s + 2·15-s + 3·19-s + 21-s − 25-s − 27-s − 8·29-s − 31-s + 33-s + 2·35-s − 37-s − 3·39-s − 6·41-s + 43-s − 2·45-s + 6·47-s − 6·49-s + 2·53-s + 2·55-s − 3·57-s − 2·59-s − 11·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.516·15-s + 0.688·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.179·31-s + 0.174·33-s + 0.338·35-s − 0.164·37-s − 0.480·39-s − 0.937·41-s + 0.152·43-s − 0.298·45-s + 0.875·47-s − 6/7·49-s + 0.274·53-s + 0.269·55-s − 0.397·57-s − 0.260·59-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.42895754528020, −13.20828753345479, −12.44084441822015, −12.07630597869482, −11.73450050844638, −11.17209826278439, −10.74162054138866, −10.45624702234646, −9.623830924251596, −9.327927317927727, −8.774505668575409, −8.065510821914754, −7.731377747091102, −7.275719441918218, −6.703589822816881, −6.133032184796366, −5.695134399138804, −5.126045755620875, −4.593188201341747, −3.860421873705296, −3.568512985220943, −3.032598379223211, −2.101236715479075, −1.473009705827199, −0.6482319874547514, 0,
0.6482319874547514, 1.473009705827199, 2.101236715479075, 3.032598379223211, 3.568512985220943, 3.860421873705296, 4.593188201341747, 5.126045755620875, 5.695134399138804, 6.133032184796366, 6.703589822816881, 7.275719441918218, 7.731377747091102, 8.065510821914754, 8.774505668575409, 9.327927317927727, 9.623830924251596, 10.45624702234646, 10.74162054138866, 11.17209826278439, 11.73450050844638, 12.07630597869482, 12.44084441822015, 13.20828753345479, 13.42895754528020
