| L(s) = 1 | + 3-s − 5-s + 3·7-s − 2·9-s + 4·11-s + 13-s − 15-s − 3·17-s + 19-s + 3·21-s − 2·23-s − 4·25-s − 5·27-s + 4·29-s + 8·31-s + 4·33-s − 3·35-s + 37-s + 39-s + 10·41-s + 5·43-s + 2·45-s + 7·47-s + 2·49-s − 3·51-s + 2·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s − 0.727·17-s + 0.229·19-s + 0.654·21-s − 0.417·23-s − 4/5·25-s − 0.962·27-s + 0.742·29-s + 1.43·31-s + 0.696·33-s − 0.507·35-s + 0.164·37-s + 0.160·39-s + 1.56·41-s + 0.762·43-s + 0.298·45-s + 1.02·47-s + 2/7·49-s − 0.420·51-s + 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.537683527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.537683527\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.318717440278510192619012802301, −8.009275093217193006626557506224, −7.12125093890743628595250217126, −6.24136212060885537165242559195, −5.52811660507401625096378789558, −4.33326634946061470771545996094, −4.08215586710840068172364753036, −2.90128267146196816290811823898, −2.04679482981860907882345217602, −0.924228925550788824117409303399,
0.924228925550788824117409303399, 2.04679482981860907882345217602, 2.90128267146196816290811823898, 4.08215586710840068172364753036, 4.33326634946061470771545996094, 5.52811660507401625096378789558, 6.24136212060885537165242559195, 7.12125093890743628595250217126, 8.009275093217193006626557506224, 8.318717440278510192619012802301
