L(s) = 1 | + (−0.846 + 1.13i)2-s − i·3-s + (−0.567 − 1.91i)4-s + (2.22 + 0.235i)5-s + (1.13 + 0.846i)6-s + (2.54 + 0.708i)7-s + (2.65 + 0.980i)8-s − 9-s + (−2.14 + 2.31i)10-s − 5.27i·11-s + (−1.91 + 0.567i)12-s − 2.60·13-s + (−2.96 + 2.28i)14-s + (0.235 − 2.22i)15-s + (−3.35 + 2.17i)16-s + 3.66·17-s + ⋯ |
L(s) = 1 | + (−0.598 + 0.801i)2-s − 0.577i·3-s + (−0.283 − 0.958i)4-s + (0.994 + 0.105i)5-s + (0.462 + 0.345i)6-s + (0.963 + 0.267i)7-s + (0.938 + 0.346i)8-s − 0.333·9-s + (−0.679 + 0.733i)10-s − 1.59i·11-s + (−0.553 + 0.163i)12-s − 0.722·13-s + (−0.791 + 0.611i)14-s + (0.0608 − 0.574i)15-s + (−0.839 + 0.544i)16-s + 0.889·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25821 - 0.0561569i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25821 - 0.0561569i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.846 - 1.13i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.22 - 0.235i)T \) |
| 7 | \( 1 + (-2.54 - 0.708i)T \) |
good | 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 - 3.66T + 17T^{2} \) |
| 19 | \( 1 + 6.51T + 19T^{2} \) |
| 23 | \( 1 - 3.54T + 23T^{2} \) |
| 29 | \( 1 - 6.64T + 29T^{2} \) |
| 31 | \( 1 - 4.51T + 31T^{2} \) |
| 37 | \( 1 + 0.593iT - 37T^{2} \) |
| 41 | \( 1 + 8.01iT - 41T^{2} \) |
| 43 | \( 1 + 0.678T + 43T^{2} \) |
| 47 | \( 1 - 5.79iT - 47T^{2} \) |
| 53 | \( 1 + 1.21iT - 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 - 0.618iT - 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 + 6.11iT - 79T^{2} \) |
| 83 | \( 1 - 12.0iT - 83T^{2} \) |
| 89 | \( 1 + 15.9iT - 89T^{2} \) |
| 97 | \( 1 - 0.241T + 97T^{2} \) |
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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
−10.87266597346835090007353607239, −10.31564809560587457585422873195, −8.987990718309364766477444552195, −8.491385626042714348785617361318, −7.55656804867469256603377282235, −6.40565919633194882122293210300, −5.76653432074315437312857263832, −4.81644931304055265825260921063, −2.56620236367955058595923943571, −1.16252746977091905685198966353,
1.61416363823686563047988232265, 2.69334020963054003079479428041, 4.47194968429097161792732940820, 4.92962663909045109820065286773, 6.67431273629450446680447027846, 7.79800930962284077166790430701, 8.714943847960366482894323507069, 9.734578860878833479571257823157, 10.16926582941992654377814107596, 10.90591108955542246538873609806