L(s) = 1 | − 2-s + 4-s + (−2.23 − 1.41i)7-s − 8-s − 1.41i·11-s + 0.926·13-s + (2.23 + 1.41i)14-s + 16-s − 2.23i·17-s + 7.63i·19-s + 1.41i·22-s − 23-s − 0.926·26-s + (−2.23 − 1.41i)28-s + 0.757i·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.845 − 0.534i)7-s − 0.353·8-s − 0.426i·11-s + 0.256·13-s + (0.597 + 0.377i)14-s + 0.250·16-s − 0.542i·17-s + 1.75i·19-s + 0.301i·22-s − 0.208·23-s − 0.181·26-s + (−0.422 − 0.267i)28-s + 0.140i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.024798768\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.024798768\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
good | 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 0.926T + 13T^{2} \) |
| 17 | \( 1 + 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 7.63iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 0.757iT - 29T^{2} \) |
| 31 | \( 1 - 4.08iT - 31T^{2} \) |
| 37 | \( 1 - 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 8.56T + 41T^{2} \) |
| 43 | \( 1 + 3.58iT - 43T^{2} \) |
| 47 | \( 1 - 1.30iT - 47T^{2} \) |
| 53 | \( 1 + 8.07T + 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 - 0.926iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 - 2.61T + 89T^{2} \) |
| 97 | \( 1 + 0.542T + 97T^{2} \) |
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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
−8.743840621064405431462734537517, −7.85771875032312643258749417121, −7.39534568754687653401770425943, −6.32627315602948006254233787377, −6.03422626147996278989405235658, −4.83334513280427940577809571266, −3.67860363391332993966766229794, −3.13311808277714411674218311228, −1.84485130205414414187123624013, −0.68926245233958698564802213843,
0.64857798360169942970717899194, 2.10663081065998814485596459061, 2.83216607829951677981930043383, 3.86657108801623563553431357814, 4.89377217948602248291522605595, 5.91519525727051990287251561004, 6.49278669314380843537135884923, 7.22824325179298796945398961774, 8.019916624890680253385440150811, 8.820522378229418430033559359743