L(s) = 1 | + (−2.84 + 2.84i)5-s − 4.61i·7-s + (1.08 − 1.08i)11-s + (−3.94 − 3.94i)13-s − 1.29·17-s + (3.08 + 3.08i)19-s − 4i·23-s − 11.2i·25-s + (−1.31 − 1.31i)29-s − 2.77·31-s + (13.1 + 13.1i)35-s + (2.81 − 2.81i)37-s + 3.03i·41-s + (−2.14 + 2.14i)43-s + 9.65·47-s + ⋯ |
L(s) = 1 | + (−1.27 + 1.27i)5-s − 1.74i·7-s + (0.326 − 0.326i)11-s + (−1.09 − 1.09i)13-s − 0.314·17-s + (0.707 + 0.707i)19-s − 0.834i·23-s − 2.24i·25-s + (−0.244 − 0.244i)29-s − 0.498·31-s + (2.22 + 2.22i)35-s + (0.462 − 0.462i)37-s + 0.473i·41-s + (−0.326 + 0.326i)43-s + 1.40·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1443497371\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1443497371\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.84 - 2.84i)T - 5iT^{2} \) |
| 7 | \( 1 + 4.61iT - 7T^{2} \) |
| 11 | \( 1 + (-1.08 + 1.08i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.94 + 3.94i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 + (-3.08 - 3.08i)T + 19iT^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + (1.31 + 1.31i)T + 29iT^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + (-2.81 + 2.81i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.03iT - 41T^{2} \) |
| 43 | \( 1 + (2.14 - 2.14i)T - 43iT^{2} \) |
| 47 | \( 1 - 9.65T + 47T^{2} \) |
| 53 | \( 1 + (6.07 - 6.07i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.05 - 4.05i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5.40 - 5.40i)T + 61iT^{2} \) |
| 67 | \( 1 + (-6.21 - 6.21i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.88iT - 71T^{2} \) |
| 73 | \( 1 + 3.50iT - 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 + (2.91 + 2.91i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.98iT - 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−8.163472890837578966210959700382, −7.71891893187194691658858879603, −7.25419550703281599616812539944, −6.73070111659391948508993833293, −5.78297137212348130811548896585, −4.56923323701986271836864306162, −3.99953995911087260392010803684, −3.33819234524061914346657123694, −2.61854554930094377539714749430, −0.916661443648784170953133551949,
0.04982099436437004862298186278, 1.55809628380726711135117712369, 2.46328676155054095007331488855, 3.55966991481317587113123512556, 4.44513783008519922805663117553, 5.06913750187316444785754465143, 5.54565730416762469750328218667, 6.74789575255390874847928508160, 7.39964931736660845971071569965, 8.168610726479472791927245126060