L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 3.41·11-s + 1.17·13-s + 16-s − 3.41·17-s + 4.82·19-s + 20-s − 3.41·22-s − 3.65·23-s + 25-s − 1.17·26-s + 5.41·29-s + 7.41·31-s − 32-s + 3.41·34-s − 3.41·37-s − 4.82·38-s − 40-s + 3.65·41-s − 7.07·43-s + 3.41·44-s + 3.65·46-s − 2.58·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.02·11-s + 0.324·13-s + 0.250·16-s − 0.828·17-s + 1.10·19-s + 0.223·20-s − 0.727·22-s − 0.762·23-s + 0.200·25-s − 0.229·26-s + 1.00·29-s + 1.33·31-s − 0.176·32-s + 0.585·34-s − 0.561·37-s − 0.783·38-s − 0.158·40-s + 0.571·41-s − 1.07·43-s + 0.514·44-s + 0.539·46-s − 0.377·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.675356483\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.675356483\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 - 0.828T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 5.89T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 0.485T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.465464335452150337620407229382, −7.76982738325878983611731111009, −6.72180655486050609785172066075, −6.49290243910954680031437989975, −5.56247320938621851542079066865, −4.62895493864919932696958207991, −3.69936007845062307469290264697, −2.74571579433716529157191727067, −1.76104732213059710084928950594, −0.845779668306668128899133781579,
0.845779668306668128899133781579, 1.76104732213059710084928950594, 2.74571579433716529157191727067, 3.69936007845062307469290264697, 4.62895493864919932696958207991, 5.56247320938621851542079066865, 6.49290243910954680031437989975, 6.72180655486050609785172066075, 7.76982738325878983611731111009, 8.465464335452150337620407229382