| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 5·11-s + 13-s + 14-s + 16-s + 17-s − 19-s + 20-s − 5·22-s + 5·23-s − 4·25-s − 26-s − 28-s + 29-s − 6·31-s − 32-s − 34-s − 35-s + 7·37-s + 38-s − 40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.50·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.229·19-s + 0.223·20-s − 1.06·22-s + 1.04·23-s − 4/5·25-s − 0.196·26-s − 0.188·28-s + 0.185·29-s − 1.07·31-s − 0.176·32-s − 0.171·34-s − 0.169·35-s + 1.15·37-s + 0.162·38-s − 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.427276231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427276231\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−9.295782986400284459761609032693, −8.873686093647145011668126099588, −7.83668502225719444226289767109, −6.95604346939463814487795088664, −6.31138561221679293740139963225, −5.55695122754200252530492708870, −4.22139674920520326615328081130, −3.31068533626162335359177819460, −2.06798281832947418473179442321, −0.970869947377069698457811585442,
0.970869947377069698457811585442, 2.06798281832947418473179442321, 3.31068533626162335359177819460, 4.22139674920520326615328081130, 5.55695122754200252530492708870, 6.31138561221679293740139963225, 6.95604346939463814487795088664, 7.83668502225719444226289767109, 8.873686093647145011668126099588, 9.295782986400284459761609032693
