L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)7-s + (1 + 1.73i)9-s + (−1.5 + 2.59i)11-s + (−1 − 3.46i)13-s + (−1.5 − 2.59i)17-s + (−2.5 − 4.33i)19-s − 0.999·21-s + (4.5 − 7.79i)23-s + 5·27-s + (4.5 − 7.79i)29-s + 8·31-s + (1.5 + 2.59i)33-s + (−3.5 + 6.06i)37-s + (−3.49 − 0.866i)39-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.188 − 0.327i)7-s + (0.333 + 0.577i)9-s + (−0.452 + 0.783i)11-s + (−0.277 − 0.960i)13-s + (−0.363 − 0.630i)17-s + (−0.573 − 0.993i)19-s − 0.218·21-s + (0.938 − 1.62i)23-s + 0.962·27-s + (0.835 − 1.44i)29-s + 1.43·31-s + (0.261 + 0.452i)33-s + (−0.575 + 0.996i)37-s + (−0.560 − 0.138i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.553182295\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.553182295\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.5 + 7.79i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.523380435122378440639863288503, −8.371630958619434782427894809532, −7.927390795834841672233041322760, −6.92814056152164990709796214236, −6.48520426433505954628962303386, −4.84473176669228677498036436346, −4.65778225521008598692859934364, −2.90752401534657191303293026800, −2.29894216460529787153784331763, −0.64709014229918454016872533216,
1.45668891779977351006651069547, 2.90831093457852372856477085870, 3.73162575624135749183917763416, 4.64326013623356427178297064586, 5.70561457074085325432182985302, 6.52510324249071274788900741352, 7.38106101876370211931141521356, 8.560708082193995969819841984458, 8.944330264298567270598688622102, 9.817978783526985625587969275345