L(s) = 1 | + (−0.707 − 0.707i)3-s − 1.73i·11-s + (1.22 + 1.22i)17-s + 1.73·19-s + (−0.707 + 0.707i)27-s + (−1.22 + 1.22i)33-s − 41-s + (−1.41 − 1.41i)43-s − i·49-s − 1.73i·51-s + (−1.22 − 1.22i)57-s + (0.707 − 0.707i)67-s + (1.22 − 1.22i)73-s + 1.00·81-s + (−0.707 − 0.707i)83-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s − 1.73i·11-s + (1.22 + 1.22i)17-s + 1.73·19-s + (−0.707 + 0.707i)27-s + (−1.22 + 1.22i)33-s − 41-s + (−1.41 − 1.41i)43-s − i·49-s − 1.73i·51-s + (−1.22 − 1.22i)57-s + (0.707 − 0.707i)67-s + (1.22 − 1.22i)73-s + 1.00·81-s + (−0.707 − 0.707i)83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9730411300\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9730411300\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 19 | \( 1 - 1.73T + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 - iT - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.444536954889937821244948426029, −7.940707056313650635670843714392, −7.03697658223884712734253880223, −6.34152358577736916953797133627, −5.60736862900162336492999890315, −5.25153758421729062678156477675, −3.55606075210030279674597227654, −3.33916659617099566238518528290, −1.66107809232308796411211818102, −0.73649380079350580759001213942,
1.35239498874749541923251808711, 2.65340595301336041708800028876, 3.63111463087516152656123284271, 4.78244133679538291342670262986, 5.00677814025596232756310214299, 5.79161395380420576832955215056, 6.94662058611434887047048979921, 7.44848887122229869708016760380, 8.175576049819130900411158563058, 9.562102801596590507835922602005