| L(s) = 1 | + (1.24 − 1.85i)5-s + 0.864·7-s + 3.90·11-s + 1.13i·13-s + 3.71·17-s + 1.72i·19-s + 9.03i·23-s + (−1.89 − 4.62i)25-s − 1.26i·29-s − 3.25i·31-s + (1.07 − 1.60i)35-s − 6.38i·37-s − 6.39i·41-s + 4.77·43-s + 4.59i·47-s + ⋯ |
| L(s) = 1 | + (0.557 − 0.830i)5-s + 0.326·7-s + 1.17·11-s + 0.314i·13-s + 0.900·17-s + 0.396i·19-s + 1.88i·23-s + (−0.379 − 0.925i)25-s − 0.235i·29-s − 0.584i·31-s + (0.182 − 0.271i)35-s − 1.05i·37-s − 0.998i·41-s + 0.728·43-s + 0.670i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.142311396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142311396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.24 + 1.85i)T \) |
| good | 7 | \( 1 - 0.864T + 7T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 - 1.13iT - 13T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 - 1.72iT - 19T^{2} \) |
| 23 | \( 1 - 9.03iT - 23T^{2} \) |
| 29 | \( 1 + 1.26iT - 29T^{2} \) |
| 31 | \( 1 + 3.25iT - 31T^{2} \) |
| 37 | \( 1 + 6.38iT - 37T^{2} \) |
| 41 | \( 1 + 6.39iT - 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 - 4.59iT - 47T^{2} \) |
| 53 | \( 1 - 8.98T + 53T^{2} \) |
| 59 | \( 1 - 8.50T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 - 4.47iT - 73T^{2} \) |
| 79 | \( 1 + 14.2iT - 79T^{2} \) |
| 83 | \( 1 - 8.10iT - 83T^{2} \) |
| 89 | \( 1 + 3.56iT - 89T^{2} \) |
| 97 | \( 1 + 10.9iT - 97T^{2} \) |
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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
−9.438462526568448154725289481506, −8.842079842429159436153864188864, −7.897577294839245900367072886212, −7.12509279308956978039764193139, −5.93603813370847464717030122964, −5.49684953412275413737675175642, −4.35164215269294555080898799928, −3.58389087837804933465636940255, −1.99770235403206181416069625680, −1.11547390209003975412316061988,
1.21486654377589016602378909101, 2.49338640682834552910745022402, 3.42417273205132072628220850410, 4.51663688586840746612871731719, 5.52784141446459770653774249970, 6.48329440130079594991744508674, 6.91887788683781961679565825217, 8.022553298768818053543714311942, 8.795257344459703359806139289415, 9.694581808363811562099717999331
