| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 3·7-s + 8-s + 9-s − 10-s + 5·11-s − 12-s − 6·13-s + 3·14-s + 15-s + 16-s − 4·17-s + 18-s + 5·19-s − 20-s − 3·21-s + 5·22-s + 5·23-s − 24-s + 25-s − 6·26-s − 27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.288·12-s − 1.66·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.14·19-s − 0.223·20-s − 0.654·21-s + 1.06·22-s + 1.04·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.223345781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223345781\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 31 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−10.25069719725221377466344362644, −9.280020980784116097299413057805, −8.280444601616760167837315202360, −7.14865555445065532452689589221, −6.82214672330097626185245453025, −5.45443905087010954704044093941, −4.74130752956692692440196387757, −4.10337014083928661947731582719, −2.64510459623110775751468942168, −1.21930154759107999167274331762,
1.21930154759107999167274331762, 2.64510459623110775751468942168, 4.10337014083928661947731582719, 4.74130752956692692440196387757, 5.45443905087010954704044093941, 6.82214672330097626185245453025, 7.14865555445065532452689589221, 8.280444601616760167837315202360, 9.280020980784116097299413057805, 10.25069719725221377466344362644
