| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 15-s + 16-s + 2·17-s − 18-s + 6·19-s + 20-s − 22-s − 24-s + 25-s + 26-s + 27-s + 2·29-s − 30-s + 4·31-s − 32-s + 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s + 0.223·20-s − 0.213·22-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.53518434405618, −12.50816879127737, −12.39443095718371, −11.85109176065943, −11.38664342474321, −10.73323563048971, −10.28653826576902, −9.926618840936150, −9.437379070838252, −9.076510959048102, −8.604528394205236, −8.049186331190570, −7.603031645163347, −7.163784572133141, −6.662240672636823, −6.151950361293999, −5.440142296688864, −5.156206378025364, −4.356816727975557, −3.777494589554239, −3.039428883636015, −2.826935544238411, −2.050480386030643, −1.389866095646250, −1.009812826470003, 0,
1.009812826470003, 1.389866095646250, 2.050480386030643, 2.826935544238411, 3.039428883636015, 3.777494589554239, 4.356816727975557, 5.156206378025364, 5.440142296688864, 6.151950361293999, 6.662240672636823, 7.163784572133141, 7.603031645163347, 8.049186331190570, 8.604528394205236, 9.076510959048102, 9.437379070838252, 9.926618840936150, 10.28653826576902, 10.73323563048971, 11.38664342474321, 11.85109176065943, 12.39443095718371, 12.50816879127737, 13.53518434405618
