| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s − 2·13-s − 15-s − 6·17-s − 4·19-s + 23-s + 25-s − 27-s − 2·29-s + 4·33-s − 2·37-s + 2·39-s + 10·41-s + 4·43-s + 45-s − 7·49-s + 6·51-s + 6·53-s − 4·55-s + 4·57-s + 4·59-s − 10·61-s − 2·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.328·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 49-s + 0.840·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s − 1.28·61-s − 0.248·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.047652508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047652508\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 23 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.072859314969051790586310563189, −7.40559907920281920505287469557, −6.60549692230683536064182234121, −6.06807619905159967066835866505, −5.16798319432792950231672210142, −4.72926718175577710065968474905, −3.80734190415831476497595963541, −2.52799243553167206193708026677, −2.07081673516810629997813322808, −0.53483081134473850571990993147,
0.53483081134473850571990993147, 2.07081673516810629997813322808, 2.52799243553167206193708026677, 3.80734190415831476497595963541, 4.72926718175577710065968474905, 5.16798319432792950231672210142, 6.06807619905159967066835866505, 6.60549692230683536064182234121, 7.40559907920281920505287469557, 8.072859314969051790586310563189
