L(s) = 1 | + 1.73i·3-s − 3.46i·7-s − 2.99·9-s + 2·13-s − 3.46i·19-s + 5.99·21-s − 5.19i·27-s − 10.3i·31-s + 10·37-s + 3.46i·39-s − 10.3i·43-s − 4.99·49-s + 5.99·57-s + 14·61-s + 10.3i·63-s + ⋯ |
L(s) = 1 | + 0.999i·3-s − 1.30i·7-s − 0.999·9-s + 0.554·13-s − 0.794i·19-s + 1.30·21-s − 0.999i·27-s − 1.86i·31-s + 1.64·37-s + 0.554i·39-s − 1.58i·43-s − 0.714·49-s + 0.794·57-s + 1.79·61-s + 1.30i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.429421661\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.429421661\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 17.3iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.806252096077230051549615030773, −9.004961524472074192758532613896, −8.109096388454995116136049978616, −7.26195777582282358024226536639, −6.27345544268238272974020158098, −5.32281012845077203772794084647, −4.23829177968066935399154452744, −3.83186428156488392870057782850, −2.55365850711334460748557892017, −0.66784532134024500454975933619,
1.33600478664384813188412092758, 2.43542790409807925484042560071, 3.36105338282589031889763531663, 4.88789315724740340779684229735, 5.87738937373433097079196097912, 6.31757887149443789052477207507, 7.39194679382306185360896595271, 8.278234613837785937109730992817, 8.762550242819512383214775020218, 9.648419446908427041658540004991