| L(s) = 1 | − 3-s + 5·7-s + 9-s − 11-s − 13-s − 3·17-s + 6·19-s − 5·21-s + 3·23-s − 27-s + 4·29-s + 33-s + 5·37-s + 39-s + 11·41-s + 6·43-s + 18·49-s + 3·51-s + 9·53-s − 6·57-s − 12·59-s + 5·61-s + 5·63-s + 8·67-s − 3·69-s − 13·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.88·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.727·17-s + 1.37·19-s − 1.09·21-s + 0.625·23-s − 0.192·27-s + 0.742·29-s + 0.174·33-s + 0.821·37-s + 0.160·39-s + 1.71·41-s + 0.914·43-s + 18/7·49-s + 0.420·51-s + 1.23·53-s − 0.794·57-s − 1.56·59-s + 0.640·61-s + 0.629·63-s + 0.977·67-s − 0.361·69-s − 1.54·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.910728352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.910728352\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−15.13794578288921, −14.50269639436867, −14.09881344586698, −13.60291058632641, −12.94143936888654, −12.29387067160111, −11.78965929938363, −11.38389879818224, −10.77578997256303, −10.64332783986716, −9.630722924241096, −9.209729202731900, −8.486871167604373, −7.878971196534319, −7.498158986136399, −6.947258863061363, −6.082655245986262, −5.462036750198485, −5.003517087462467, −4.498876327813326, −3.935779529560526, −2.769725764594552, −2.242079171376778, −1.273371430108238, −0.7697808345624478,
0.7697808345624478, 1.273371430108238, 2.242079171376778, 2.769725764594552, 3.935779529560526, 4.498876327813326, 5.003517087462467, 5.462036750198485, 6.082655245986262, 6.947258863061363, 7.498158986136399, 7.878971196534319, 8.486871167604373, 9.209729202731900, 9.630722924241096, 10.64332783986716, 10.77578997256303, 11.38389879818224, 11.78965929938363, 12.29387067160111, 12.94143936888654, 13.60291058632641, 14.09881344586698, 14.50269639436867, 15.13794578288921
