| L(s) = 1 | + 2·3-s − 5-s + 9-s − 11-s − 2·15-s − 2·17-s + 4·19-s − 4·23-s + 25-s − 4·27-s − 6·29-s − 6·31-s − 2·33-s − 6·37-s − 4·41-s − 10·43-s − 45-s − 8·47-s − 7·49-s − 4·51-s + 4·53-s + 55-s + 8·57-s − 8·59-s + 14·61-s − 4·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 1.07·31-s − 0.348·33-s − 0.986·37-s − 0.624·41-s − 1.52·43-s − 0.149·45-s − 1.16·47-s − 49-s − 0.560·51-s + 0.549·53-s + 0.134·55-s + 1.05·57-s − 1.04·59-s + 1.79·61-s − 0.488·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.330074674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.330074674\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.19819559750230, −13.44451806882700, −13.27797461704763, −12.79031780511910, −11.93924629008818, −11.65700032635964, −11.16716940468248, −10.44195781698730, −9.977778594773768, −9.395233057938286, −8.996619310064129, −8.380987084145434, −8.067907806975914, −7.488782516012330, −7.061512550510188, −6.417352841555364, −5.600938625011173, −5.183305510647039, −4.459417566546696, −3.715894428475258, −3.371506238524357, −2.882140375288398, −1.869677627245022, −1.743580499054449, −0.3249665963388361,
0.3249665963388361, 1.743580499054449, 1.869677627245022, 2.882140375288398, 3.371506238524357, 3.715894428475258, 4.459417566546696, 5.183305510647039, 5.600938625011173, 6.417352841555364, 7.061512550510188, 7.488782516012330, 8.067907806975914, 8.380987084145434, 8.996619310064129, 9.395233057938286, 9.977778594773768, 10.44195781698730, 11.16716940468248, 11.65700032635964, 11.93924629008818, 12.79031780511910, 13.27797461704763, 13.44451806882700, 14.19819559750230
