| L(s) = 1 | + 3-s − 3·7-s + 9-s + 3·11-s − 13-s − 3·17-s − 3·21-s − 3·23-s + 27-s + 8·29-s + 4·31-s + 3·33-s − 37-s − 39-s − 3·41-s + 4·43-s + 10·47-s + 2·49-s − 3·51-s + 9·53-s + 4·59-s + 9·61-s − 3·63-s − 4·67-s − 3·69-s + 7·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.727·17-s − 0.654·21-s − 0.625·23-s + 0.192·27-s + 1.48·29-s + 0.718·31-s + 0.522·33-s − 0.164·37-s − 0.160·39-s − 0.468·41-s + 0.609·43-s + 1.45·47-s + 2/7·49-s − 0.420·51-s + 1.23·53-s + 0.520·59-s + 1.15·61-s − 0.377·63-s − 0.488·67-s − 0.361·69-s + 0.830·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.038702550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.038702550\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−8.676293579990799815272638597740, −7.77114296691181890663684370718, −6.79291391123096415696792688214, −6.55498303713232398677862848184, −5.58813170330544494387430900852, −4.43800688964065233511637925240, −3.85193896233857651410877078922, −2.94428986411390513297348260759, −2.17606181274358899935371352046, −0.793721380106708964001341450655,
0.793721380106708964001341450655, 2.17606181274358899935371352046, 2.94428986411390513297348260759, 3.85193896233857651410877078922, 4.43800688964065233511637925240, 5.58813170330544494387430900852, 6.55498303713232398677862848184, 6.79291391123096415696792688214, 7.77114296691181890663684370718, 8.676293579990799815272638597740
