| L(s) = 1 | + 2·3-s + 7-s + 9-s − 11-s + 2·13-s − 17-s − 5·19-s + 2·21-s + 6·23-s − 4·27-s − 5·29-s − 7·31-s − 2·33-s + 6·37-s + 4·39-s + 9·41-s − 2·43-s + 49-s − 2·51-s − 10·57-s + 7·59-s − 4·61-s + 63-s − 4·67-s + 12·69-s + 8·71-s − 11·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.242·17-s − 1.14·19-s + 0.436·21-s + 1.25·23-s − 0.769·27-s − 0.928·29-s − 1.25·31-s − 0.348·33-s + 0.986·37-s + 0.640·39-s + 1.40·41-s − 0.304·43-s + 1/7·49-s − 0.280·51-s − 1.32·57-s + 0.911·59-s − 0.512·61-s + 0.125·63-s − 0.488·67-s + 1.44·69-s + 0.949·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.228484721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.228484721\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.10725952462777, −12.93907478668131, −12.30740429951610, −11.58668125229179, −11.02609794882571, −10.94696056750877, −10.32303064532213, −9.557916972710061, −9.221281028834902, −8.828674582996748, −8.428971416051224, −7.853400969824238, −7.537267900867742, −6.967255977862178, −6.369650456847197, −5.746550983091205, −5.337155583850162, −4.605970238363818, −4.025290529062381, −3.709164735269161, −2.907721235645369, −2.570661495726943, −1.922974898477871, −1.392687774347264, −0.4524192347391059,
0.4524192347391059, 1.392687774347264, 1.922974898477871, 2.570661495726943, 2.907721235645369, 3.709164735269161, 4.025290529062381, 4.605970238363818, 5.337155583850162, 5.746550983091205, 6.369650456847197, 6.967255977862178, 7.537267900867742, 7.853400969824238, 8.428971416051224, 8.828674582996748, 9.221281028834902, 9.557916972710061, 10.32303064532213, 10.94696056750877, 11.02609794882571, 11.58668125229179, 12.30740429951610, 12.93907478668131, 13.10725952462777
