L(s) = 1 | + (−1 + 1.73i)5-s + (2.5 − 0.866i)7-s + (1 + 1.73i)11-s + 13-s + (0.5 − 0.866i)19-s + (0.500 + 0.866i)25-s − 4·29-s + (4.5 + 7.79i)31-s + (−1.00 + 5.19i)35-s + (−1.5 + 2.59i)37-s + 10·41-s − 5·43-s + (3 − 5.19i)47-s + (5.5 − 4.33i)49-s + (6 + 10.3i)53-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s + (0.944 − 0.327i)7-s + (0.301 + 0.522i)11-s + 0.277·13-s + (0.114 − 0.198i)19-s + (0.100 + 0.173i)25-s − 0.742·29-s + (0.808 + 1.39i)31-s + (−0.169 + 0.878i)35-s + (−0.246 + 0.427i)37-s + 1.56·41-s − 0.762·43-s + (0.437 − 0.757i)47-s + (0.785 − 0.618i)49-s + (0.824 + 1.42i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.629895187\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.629895187\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-8 + 13.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.36399762250120573814598708069, −9.182747633313385139873162195338, −8.387715267019171415536021809444, −7.39158775256586575644915074581, −7.00618916862623313161374910518, −5.80721643781279534661433125241, −4.72476117081215116041799305512, −3.88893713792795146375130975510, −2.75164723885779537649314173473, −1.37038317223187188499223624872,
0.863655527673210234180751128925, 2.21317402520424151676837107087, 3.71501820205544120546810516918, 4.56439880910610383096070532036, 5.44715015676292965681684339627, 6.31955252794123998163119977718, 7.61856785324153597856786510668, 8.204672336535006642549671801384, 8.884364128995608565758699400245, 9.689550786027012030029865932036