L(s) = 1 | − 2.82i·5-s − 1.41i·7-s + 2·13-s − 1.41i·17-s − 2.82i·19-s + 6·23-s − 3.00·25-s + 5.65i·29-s − 9.89i·31-s − 4.00·35-s − 2·37-s + 4.24i·41-s − 8.48i·43-s + 10·47-s + 5·49-s + ⋯ |
L(s) = 1 | − 1.26i·5-s − 0.534i·7-s + 0.554·13-s − 0.342i·17-s − 0.648i·19-s + 1.25·23-s − 0.600·25-s + 1.05i·29-s − 1.77i·31-s − 0.676·35-s − 0.328·37-s + 0.662i·41-s − 1.29i·43-s + 1.45·47-s + 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.804818769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804818769\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 9.89iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 12.7iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.151184771621422146874822353018, −7.34189631075212151148084317252, −6.73455857713300195719825659145, −5.68564825947825612859627414756, −5.09075383026184885821874152957, −4.37908441592510775091517185739, −3.63990661946272883795797310598, −2.53062676954705139837227007184, −1.30562994308230449742493869779, −0.55307771627804534081808136931,
1.30307726128232653728848458857, 2.46825761757428208425754601203, 3.12083841645615103625805546097, 3.88778964315032066822358969136, 4.91944239709002797886671611082, 5.86655015292593048227883369808, 6.33330154045577279542136862733, 7.14378782845067483818111445199, 7.69000874155822862744733328026, 8.708939325123522326294394013957