L(s) = 1 | + (0.622 + 1.26i)2-s + (−2.09 + 2.09i)3-s + (−1.22 + 1.58i)4-s + (−3.96 − 1.35i)6-s + (0.707 + 0.707i)7-s + (−2.76 − 0.572i)8-s − 5.78i·9-s − 0.214i·11-s + (−0.744 − 5.88i)12-s + (−4.29 − 4.29i)13-s + (−0.457 + 1.33i)14-s + (−0.996 − 3.87i)16-s + (−2.00 + 2.00i)17-s + (7.34 − 3.60i)18-s − 0.877·19-s + ⋯ |
L(s) = 1 | + (0.440 + 0.897i)2-s + (−1.21 + 1.21i)3-s + (−0.612 + 0.790i)4-s + (−1.61 − 0.554i)6-s + (0.267 + 0.267i)7-s + (−0.979 − 0.202i)8-s − 1.92i·9-s − 0.0648i·11-s + (−0.214 − 1.69i)12-s + (−1.19 − 1.19i)13-s + (−0.122 + 0.357i)14-s + (−0.249 − 0.968i)16-s + (−0.487 + 0.487i)17-s + (1.73 − 0.848i)18-s − 0.201·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0522855 - 0.0249763i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0522855 - 0.0249763i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.622 - 1.26i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (2.09 - 2.09i)T - 3iT^{2} \) |
| 11 | \( 1 + 0.214iT - 11T^{2} \) |
| 13 | \( 1 + (4.29 + 4.29i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.00 - 2.00i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.877T + 19T^{2} \) |
| 23 | \( 1 + (0.0902 - 0.0902i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.03iT - 29T^{2} \) |
| 31 | \( 1 - 8.60iT - 31T^{2} \) |
| 37 | \( 1 + (-1.29 + 1.29i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 + (2.06 - 2.06i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.88 + 4.88i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.77 - 2.77i)T + 53iT^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 8.32T + 61T^{2} \) |
| 67 | \( 1 + (0.555 + 0.555i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.75iT - 71T^{2} \) |
| 73 | \( 1 + (4.18 + 4.18i)T + 73iT^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + (-5.30 + 5.30i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.7iT - 89T^{2} \) |
| 97 | \( 1 + (-2.58 + 2.58i)T - 97iT^{2} \) |
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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
−11.11037396360229145172233389198, −10.31655985240910606357563569024, −9.552816804458739201398033511153, −8.592518695485613254540510396061, −7.57668311356499326314110202280, −6.45560487140030395277878676035, −5.66217978324284152369938676252, −4.97672938386598329929266291335, −4.30056859145328635128329157272, −3.06743035188369678627209021341,
0.03059486263527961543714318555, 1.51621547852878081158542555743, 2.48652827553978995639038183885, 4.34343222207263778616745556200, 5.03861181240412676251782621718, 6.07872537519347773693776131417, 6.87294207287652687709491120563, 7.70500221814344123302924906933, 9.074435028288306827647202825666, 9.983993078650899872967408339761