L(s) = 1 | − 5.26·5-s + 1.82·7-s + 15.7·11-s − 14.8i·13-s + 17.7·17-s + (−10.9 − 15.5i)19-s + 44.0·23-s + 2.72·25-s + 0.665i·29-s + 45.5i·31-s − 9.58·35-s + 7.42i·37-s + 55.1i·41-s − 57.7·43-s − 59.0·47-s + ⋯ |
L(s) = 1 | − 1.05·5-s + 0.260·7-s + 1.42·11-s − 1.14i·13-s + 1.04·17-s + (−0.574 − 0.818i)19-s + 1.91·23-s + 0.108·25-s + 0.0229i·29-s + 1.47i·31-s − 0.273·35-s + 0.200i·37-s + 1.34i·41-s − 1.34·43-s − 1.25·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.827706272\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.827706272\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (10.9 + 15.5i)T \) |
good | 5 | \( 1 + 5.26T + 25T^{2} \) |
| 7 | \( 1 - 1.82T + 49T^{2} \) |
| 11 | \( 1 - 15.7T + 121T^{2} \) |
| 13 | \( 1 + 14.8iT - 169T^{2} \) |
| 17 | \( 1 - 17.7T + 289T^{2} \) |
| 23 | \( 1 - 44.0T + 529T^{2} \) |
| 29 | \( 1 - 0.665iT - 841T^{2} \) |
| 31 | \( 1 - 45.5iT - 961T^{2} \) |
| 37 | \( 1 - 7.42iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 55.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 57.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 59.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 45.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 98.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 8.94iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 39.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 42.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 91.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 40.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 20.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 120. iT - 9.40e3T^{2} \) |
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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
−8.347399839480816691486824419876, −7.964455516183405016081428057598, −6.91315409376556714809040445377, −6.52428363805928965153954979191, −5.16482132828864906178197289803, −4.72297338695116121069151480612, −3.45579844632931321461441569807, −3.20687338790189254453385018429, −1.50463502280880051430971119284, −0.56330080915936538741634323282,
0.925668243894936368193744761475, 1.91420393054056952824151588381, 3.36930449154754037378956253799, 3.96057620156203338740038140357, 4.63715707331309046871702167060, 5.71503641619176529875932271622, 6.66053690083042275384024207377, 7.20676330305250158938396385862, 8.031353499905733060396829559361, 8.732147164316173641144131297088