L(s) = 1 | + (−2.13 + 3.70i)5-s + (1.5 + 2.17i)7-s + (2.13 + 3.70i)11-s − 1.27·13-s + (−2 − 3.46i)17-s + (0.637 − 1.10i)19-s + (−2 + 3.46i)23-s + (−6.63 − 11.4i)25-s + 2.27·29-s + (−0.5 − 0.866i)31-s + (−11.2 + 0.894i)35-s + (−2.63 + 4.56i)37-s − 10.5·41-s + 7.27·43-s + (3 − 5.19i)47-s + ⋯ |
L(s) = 1 | + (−0.955 + 1.65i)5-s + (0.566 + 0.823i)7-s + (0.644 + 1.11i)11-s − 0.353·13-s + (−0.485 − 0.840i)17-s + (0.146 − 0.253i)19-s + (−0.417 + 0.722i)23-s + (−1.32 − 2.29i)25-s + 0.422·29-s + (−0.0898 − 0.155i)31-s + (−1.90 + 0.151i)35-s + (−0.433 + 0.751i)37-s − 1.64·41-s + 1.10·43-s + (0.437 − 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.022798142\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.022798142\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.5 - 2.17i)T \) |
good | 5 | \( 1 + (2.13 - 3.70i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.13 - 3.70i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.637 + 1.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.27T + 29T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.63 - 4.56i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.862 - 1.49i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.13 + 5.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.63 - 6.30i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (1.63 + 2.83i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.77 - 3.07i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 0.274T + 83T^{2} \) |
| 89 | \( 1 + (-2.27 + 3.94i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−10.36523570406891729770480222141, −9.607400063138499689527242807207, −8.609519278011142609806469146202, −7.58347849229296084307954943277, −7.10080134979578139985976132335, −6.31525346625050145208014304174, −5.02325002005433665857439083871, −4.07459296531046971612184733042, −2.98406566647285375159988252631, −2.08441138762200921795235015324,
0.47990223343305405332636167682, 1.56047418139984074769949336161, 3.60588585897571327879094872357, 4.26400889129282666156897449812, 5.01966805967998860028761126360, 6.09326389955812041634778669473, 7.28891576527777033059573531737, 8.197077900292418700930136963777, 8.545826658739662259428343321006, 9.375133011589056443898460150463