| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 4·11-s − 12-s + 13-s + 16-s + 2·17-s + 18-s + 8·19-s − 4·22-s − 24-s + 26-s − 27-s + 6·29-s + 4·31-s + 32-s + 4·33-s + 2·34-s + 36-s + 2·37-s + 8·38-s − 39-s + 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.83·19-s − 0.852·22-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.696·33-s + 0.342·34-s + 1/6·36-s + 0.328·37-s + 1.29·38-s − 0.160·39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.303738050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.303738050\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−13.78868614451259, −13.34834054639176, −12.79497177352420, −12.34835201736052, −11.85256864529930, −11.51977427993967, −10.88181561516748, −10.44750252114413, −10.00205726677088, −9.512866620133999, −8.808020146173401, −8.012966062010215, −7.759498936167462, −7.171101627572893, −6.646047641605216, −5.931949669511901, −5.578999545221876, −5.099423894412278, −4.622872460542820, −3.934562322811375, −3.273925743858084, −2.731285974811732, −2.173433194143654, −1.066281348498612, −0.7220395176803354,
0.7220395176803354, 1.066281348498612, 2.173433194143654, 2.731285974811732, 3.273925743858084, 3.934562322811375, 4.622872460542820, 5.099423894412278, 5.578999545221876, 5.931949669511901, 6.646047641605216, 7.171101627572893, 7.759498936167462, 8.012966062010215, 8.808020146173401, 9.512866620133999, 10.00205726677088, 10.44750252114413, 10.88181561516748, 11.51977427993967, 11.85256864529930, 12.34835201736052, 12.79497177352420, 13.34834054639176, 13.78868614451259
