| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 2·11-s + 4·14-s + 16-s + 2·17-s − 6·19-s + 2·22-s + 6·23-s − 4·28-s − 2·29-s + 6·31-s − 32-s − 2·34-s − 2·37-s + 6·38-s + 10·41-s + 10·43-s − 2·44-s − 6·46-s + 12·47-s + 9·49-s + 2·53-s + 4·56-s + 2·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 0.603·11-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 1.37·19-s + 0.426·22-s + 1.25·23-s − 0.755·28-s − 0.371·29-s + 1.07·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.973·38-s + 1.56·41-s + 1.52·43-s − 0.301·44-s − 0.884·46-s + 1.75·47-s + 9/7·49-s + 0.274·53-s + 0.534·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265864046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265864046\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.99981189826948, −13.46069203018709, −12.90447693321566, −12.51004532869321, −12.31076016151851, −11.33605049346806, −10.97034117057906, −10.37372496424163, −10.09935760521403, −9.476385281760709, −8.987428251802548, −8.656621910739655, −7.910973153661526, −7.345830706543413, −6.953124082509343, −6.311686952398488, −5.902981046545039, −5.359331937151470, −4.441785128172073, −3.901629445985014, −3.170691266599008, −2.597457311484171, −2.214833281448687, −0.9640518231558059, −0.5088488503517972,
0.5088488503517972, 0.9640518231558059, 2.214833281448687, 2.597457311484171, 3.170691266599008, 3.901629445985014, 4.441785128172073, 5.359331937151470, 5.902981046545039, 6.311686952398488, 6.953124082509343, 7.345830706543413, 7.910973153661526, 8.656621910739655, 8.987428251802548, 9.476385281760709, 10.09935760521403, 10.37372496424163, 10.97034117057906, 11.33605049346806, 12.31076016151851, 12.51004532869321, 12.90447693321566, 13.46069203018709, 13.99981189826948
