| L(s) = 1 | − 4·7-s − 13-s − 2·17-s − 4·19-s + 8·23-s + 2·29-s − 8·31-s + 2·37-s + 6·41-s + 12·43-s + 9·49-s − 10·53-s + 10·61-s − 4·67-s + 16·71-s + 6·73-s − 8·79-s + 4·83-s + 14·89-s + 4·91-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.277·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 0.371·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s + 1.82·43-s + 9/7·49-s − 1.37·53-s + 1.28·61-s − 0.488·67-s + 1.89·71-s + 0.702·73-s − 0.900·79-s + 0.439·83-s + 1.48·89-s + 0.419·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.626294097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.626294097\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| 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
−12.96392391465145, −12.72721388261281, −12.47881789136947, −11.71623292881244, −11.10568429596097, −10.75532582007855, −10.43393205995074, −9.547149586388307, −9.423994423080337, −9.023290842026323, −8.457332332676607, −7.769935260614643, −7.255008426012401, −6.797813508383054, −6.411495503528408, −5.896729117303167, −5.360257148704156, −4.688317634972228, −4.196768501548876, −3.538127430896252, −3.136928435556612, −2.456081083926565, −2.057196521922973, −0.9542586684373653, −0.4299141709658364,
0.4299141709658364, 0.9542586684373653, 2.057196521922973, 2.456081083926565, 3.136928435556612, 3.538127430896252, 4.196768501548876, 4.688317634972228, 5.360257148704156, 5.896729117303167, 6.411495503528408, 6.797813508383054, 7.255008426012401, 7.769935260614643, 8.457332332676607, 9.023290842026323, 9.423994423080337, 9.547149586388307, 10.43393205995074, 10.75532582007855, 11.10568429596097, 11.71623292881244, 12.47881789136947, 12.72721388261281, 12.96392391465145
