| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s − 3·9-s + 10-s + 6·11-s + 2·13-s + 16-s − 17-s + 3·18-s + 8·19-s − 20-s − 6·22-s − 4·23-s + 25-s − 2·26-s + 10·29-s + 8·31-s − 32-s + 34-s − 3·36-s − 8·37-s − 8·38-s + 40-s + 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 9-s + 0.316·10-s + 1.80·11-s + 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.707·18-s + 1.83·19-s − 0.223·20-s − 1.27·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s + 1.85·29-s + 1.43·31-s − 0.176·32-s + 0.171·34-s − 1/2·36-s − 1.31·37-s − 1.29·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533331093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533331093\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−7.968956062762538663264856994410, −7.13152010320404877000395086042, −6.46799927611923595864478455072, −6.01378114894445963131008474598, −5.05354943766348752376202857243, −4.13034705332807197479813258127, −3.37786152082892232946548178321, −2.71551189403222211564421296461, −1.44555906992270630072636016668, −0.74549733914556493987755618115,
0.74549733914556493987755618115, 1.44555906992270630072636016668, 2.71551189403222211564421296461, 3.37786152082892232946548178321, 4.13034705332807197479813258127, 5.05354943766348752376202857243, 6.01378114894445963131008474598, 6.46799927611923595864478455072, 7.13152010320404877000395086042, 7.968956062762538663264856994410
