L(s) = 1 | + (−0.5 + 0.866i)5-s + (−2.5 + 0.866i)7-s + (−2 − 3.46i)11-s + 7·13-s + (−3 − 5.19i)17-s + (−1.5 + 2.59i)19-s + (−1 + 1.73i)23-s + (−0.499 − 0.866i)25-s + 2·29-s + (−3.5 − 6.06i)31-s + (0.500 − 2.59i)35-s + (3.5 − 6.06i)37-s + 8·41-s + 5·43-s + (5 − 8.66i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (−0.944 + 0.327i)7-s + (−0.603 − 1.04i)11-s + 1.94·13-s + (−0.727 − 1.26i)17-s + (−0.344 + 0.596i)19-s + (−0.208 + 0.361i)23-s + (−0.0999 − 0.173i)25-s + 0.371·29-s + (−0.628 − 1.08i)31-s + (0.0845 − 0.439i)35-s + (0.575 − 0.996i)37-s + 1.24·41-s + 0.762·43-s + (0.729 − 1.26i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.123385055\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.123385055\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 7T + 13T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4 + 6.92i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (7.5 + 12.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.394532182480944874267188999577, −8.800821680521949118612347702871, −7.971589575672985680435574219341, −7.01161671093974062175757734310, −6.03141849305104832464210902560, −5.70351554180458461424543216130, −4.09285998962856956311189882601, −3.37205058030186558840454043625, −2.39985290215922201020112336055, −0.52099052843588142411081167438,
1.25171298357288026216613093534, 2.69372013546873908501875540809, 3.90886557178600735210015007608, 4.47587220652628025851132830824, 5.84890174401273701277718597407, 6.46413072487416308669651032960, 7.34947147632109602011473184340, 8.379815792294676150069973695893, 8.915855268582601702154142891928, 9.854841928857544441656671985006