| L(s) = 1 | + 2-s + 4-s − 2·5-s − 2·7-s + 8-s − 2·10-s + 4·11-s + 4·13-s − 2·14-s + 16-s − 2·20-s + 4·22-s + 4·23-s − 25-s + 4·26-s − 2·28-s − 4·31-s + 32-s + 4·35-s − 2·37-s − 2·40-s − 41-s − 12·43-s + 4·44-s + 4·46-s + 2·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s + 0.784·26-s − 0.377·28-s − 0.718·31-s + 0.176·32-s + 0.676·35-s − 0.328·37-s − 0.316·40-s − 0.156·41-s − 1.82·43-s + 0.603·44-s + 0.589·46-s + 0.291·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 17 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.09790878277820, −12.97091191585084, −12.12673332828955, −11.92438477882185, −11.55796677028303, −10.97303747736604, −10.64549241481906, −10.01168116993300, −9.415316940409772, −8.968448996347875, −8.541178161209212, −7.986565473686541, −7.355741270379129, −6.953731925173168, −6.473796464505379, −6.079223729820337, −5.557191518047780, −4.791729194447321, −4.401805634785400, −3.720233653070193, −3.399257205090486, −3.189003443784965, −2.132389223997818, −1.530885667035695, −0.8635468570967088, 0,
0.8635468570967088, 1.530885667035695, 2.132389223997818, 3.189003443784965, 3.399257205090486, 3.720233653070193, 4.401805634785400, 4.791729194447321, 5.557191518047780, 6.079223729820337, 6.473796464505379, 6.953731925173168, 7.355741270379129, 7.986565473686541, 8.541178161209212, 8.968448996347875, 9.415316940409772, 10.01168116993300, 10.64549241481906, 10.97303747736604, 11.55796677028303, 11.92438477882185, 12.12673332828955, 12.97091191585084, 13.09790878277820
