L(s) = 1 | + 3·9-s − 6i·13-s + 2i·17-s + 10·29-s − 2i·37-s + 10·41-s + 7·49-s − 14i·53-s − 10·61-s + 6i·73-s + 9·81-s − 10·89-s + 18i·97-s − 2·101-s − 6·109-s + ⋯ |
L(s) = 1 | + 9-s − 1.66i·13-s + 0.485i·17-s + 1.85·29-s − 0.328i·37-s + 1.56·41-s + 49-s − 1.92i·53-s − 1.28·61-s + 0.702i·73-s + 81-s − 1.05·89-s + 1.82i·97-s − 0.199·101-s − 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61397 - 0.381007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61397 - 0.381007i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−10.33520937171866579085045591813, −9.480292352571468861477527396633, −8.364033683732178246378748509922, −7.72030003281674611875320911196, −6.75814582302984525899496199797, −5.78985949289723735257272644440, −4.80272843830423347394549676359, −3.76504169895643042056289074653, −2.58961987590985827320755071125, −1.00561549055021288502486662352,
1.34455815818838023794816884635, 2.66520218597480113038850523565, 4.16368315818889048506434704127, 4.68213637693609946801652378695, 6.09645881279135912507132147456, 6.90485813285815305524578174386, 7.61372460765992789645661953803, 8.802311016877131148333845890323, 9.451278719798306226364636225712, 10.27707991207924662442301518911