| L(s) = 1 | − 4·7-s − 3·9-s + 6·11-s + 2·13-s + 6·17-s − 6·19-s − 23-s − 5·25-s + 6·29-s + 8·37-s + 6·41-s − 2·43-s + 8·47-s + 9·49-s + 8·53-s + 4·59-s + 4·61-s + 12·63-s + 2·67-s + 8·71-s + 6·73-s − 24·77-s − 12·79-s + 9·81-s + 10·83-s + 10·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 9-s + 1.80·11-s + 0.554·13-s + 1.45·17-s − 1.37·19-s − 0.208·23-s − 25-s + 1.11·29-s + 1.31·37-s + 0.937·41-s − 0.304·43-s + 1.16·47-s + 9/7·49-s + 1.09·53-s + 0.520·59-s + 0.512·61-s + 1.51·63-s + 0.244·67-s + 0.949·71-s + 0.702·73-s − 2.73·77-s − 1.35·79-s + 81-s + 1.09·83-s + 1.05·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.410709723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410709723\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−9.482275041588542076858354761831, −8.819349485473852491025989896278, −8.036767647289765612325445192469, −6.81837654905300671653257532467, −6.20342556021711795736270464252, −5.74191918395678311249301846034, −4.09996527715525771634961973849, −3.56765835233795064902721464962, −2.50382974127178624605886452113, −0.851509710399297621880365802324,
0.851509710399297621880365802324, 2.50382974127178624605886452113, 3.56765835233795064902721464962, 4.09996527715525771634961973849, 5.74191918395678311249301846034, 6.20342556021711795736270464252, 6.81837654905300671653257532467, 8.036767647289765612325445192469, 8.819349485473852491025989896278, 9.482275041588542076858354761831
