| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 2·12-s + 16-s − 6·17-s − 18-s + 2·24-s + 4·27-s + 6·29-s − 6·31-s − 32-s + 6·34-s + 36-s − 6·37-s − 10·43-s + 12·47-s − 2·48-s − 7·49-s + 12·51-s − 4·54-s − 6·58-s − 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.577·12-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.408·24-s + 0.769·27-s + 1.11·29-s − 1.07·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.986·37-s − 1.52·43-s + 1.75·47-s − 0.288·48-s − 49-s + 1.68·51-s − 0.544·54-s − 0.787·58-s − 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4330084323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4330084323\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−7.75613950210011250344406542843, −6.90184759671754926414253538065, −6.55889408141557873194856992059, −5.84693920213512216759559897487, −5.10241202684064360607024042961, −4.48052475065147540071218448359, −3.43176869510151927907729223598, −2.44360615240306294039830653198, −1.52099788743189649539218248765, −0.38606949228854182018999113161,
0.38606949228854182018999113161, 1.52099788743189649539218248765, 2.44360615240306294039830653198, 3.43176869510151927907729223598, 4.48052475065147540071218448359, 5.10241202684064360607024042961, 5.84693920213512216759559897487, 6.55889408141557873194856992059, 6.90184759671754926414253538065, 7.75613950210011250344406542843
