| L(s) = 1 | − 2·7-s + 11-s − 6·13-s − 4·17-s − 2·19-s − 8·23-s + 6·37-s − 10·43-s − 3·49-s + 14·53-s + 12·59-s − 14·61-s − 4·67-s − 6·73-s − 2·77-s + 2·79-s + 16·83-s + 14·89-s + 12·91-s + 2·97-s − 8·101-s + 12·103-s + 16·107-s − 18·109-s − 10·113-s + 8·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s − 1.66·13-s − 0.970·17-s − 0.458·19-s − 1.66·23-s + 0.986·37-s − 1.52·43-s − 3/7·49-s + 1.92·53-s + 1.56·59-s − 1.79·61-s − 0.488·67-s − 0.702·73-s − 0.227·77-s + 0.225·79-s + 1.75·83-s + 1.48·89-s + 1.25·91-s + 0.203·97-s − 0.796·101-s + 1.18·103-s + 1.54·107-s − 1.72·109-s − 0.940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8223772765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8223772765\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.62489181184421234453131291890, −6.90157192143013503315664721378, −6.40028968233081218344089186517, −5.70973916640614106852281535104, −4.79609870166504673319333599431, −4.24768922780384411028798986081, −3.42641140099800615964524694297, −2.48287779496817200182225801565, −1.94583211918079216139620586299, −0.40159584923319132486875397972,
0.40159584923319132486875397972, 1.94583211918079216139620586299, 2.48287779496817200182225801565, 3.42641140099800615964524694297, 4.24768922780384411028798986081, 4.79609870166504673319333599431, 5.70973916640614106852281535104, 6.40028968233081218344089186517, 6.90157192143013503315664721378, 7.62489181184421234453131291890
