| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 3·7-s − 3·8-s + 9-s − 2·11-s + 12-s + 3·14-s − 16-s + 4·17-s + 18-s + 19-s − 3·21-s − 2·22-s + 7·23-s + 3·24-s − 27-s − 3·28-s − 8·29-s + 31-s + 5·32-s + 2·33-s + 4·34-s − 36-s + 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.13·7-s − 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 0.801·14-s − 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.229·19-s − 0.654·21-s − 0.426·22-s + 1.45·23-s + 0.612·24-s − 0.192·27-s − 0.566·28-s − 1.48·29-s + 0.179·31-s + 0.883·32-s + 0.348·33-s + 0.685·34-s − 1/6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.806009714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.806009714\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−12.93521942531625, −12.46635239630201, −12.00516267841766, −11.55845531645839, −11.14073754844449, −10.72152458430857, −10.16936449092800, −9.667201196515128, −9.085711489981751, −8.784380822864644, −8.045796884478376, −7.680948246837233, −7.301920028953496, −6.565135928165261, −5.931214724404917, −5.589103186693355, −5.002087239483645, −4.888149213448432, −4.305573433647417, −3.563597387270157, −3.239970308889463, −2.477886593519865, −1.769718378896353, −1.058753891289462, −0.4886428451948017,
0.4886428451948017, 1.058753891289462, 1.769718378896353, 2.477886593519865, 3.239970308889463, 3.563597387270157, 4.305573433647417, 4.888149213448432, 5.002087239483645, 5.589103186693355, 5.931214724404917, 6.565135928165261, 7.301920028953496, 7.680948246837233, 8.045796884478376, 8.784380822864644, 9.085711489981751, 9.667201196515128, 10.16936449092800, 10.72152458430857, 11.14073754844449, 11.55845531645839, 12.00516267841766, 12.46635239630201, 12.93521942531625
