L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.521 + 0.300i)5-s + (−0.499 − 0.866i)9-s + (0.141 + 0.0818i)11-s + 3.37i·13-s + 0.601i·15-s + (1.84 + 1.06i)17-s + (0.476 + 0.825i)19-s + (−6.05 + 3.49i)23-s + (−2.31 + 4.01i)25-s − 0.999·27-s − 1.28·29-s + (1.67 − 2.90i)31-s + (0.141 − 0.0818i)33-s + (−2.58 − 4.47i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.233 + 0.134i)5-s + (−0.166 − 0.288i)9-s + (0.0427 + 0.0246i)11-s + 0.937i·13-s + 0.155i·15-s + (0.447 + 0.258i)17-s + (0.109 + 0.189i)19-s + (−1.26 + 0.729i)23-s + (−0.463 + 0.803i)25-s − 0.192·27-s − 0.239·29-s + (0.301 − 0.521i)31-s + (0.0246 − 0.0142i)33-s + (−0.424 − 0.735i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.255557425\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.255557425\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.521 - 0.300i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.141 - 0.0818i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 + (-1.84 - 1.06i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.476 - 0.825i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.05 - 3.49i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.28T + 29T^{2} \) |
| 31 | \( 1 + (-1.67 + 2.90i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.58 + 4.47i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.67iT - 41T^{2} \) |
| 43 | \( 1 - 6.09iT - 43T^{2} \) |
| 47 | \( 1 + (1.84 + 3.20i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.94 - 12.0i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.02 + 6.97i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.157 + 0.0907i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.87 - 4.54i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.36iT - 71T^{2} \) |
| 73 | \( 1 + (-10.5 - 6.07i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.05 - 3.49i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + (12.8 - 7.44i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.08iT - 97T^{2} \) |
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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
−9.324838370662699813653587103475, −8.084205303457448238484104062979, −7.81492626291808972567781008323, −6.86366760943640782701205959968, −6.16542470455332050288407033697, −5.31614011671942140236990122241, −4.13325538410675445568931421790, −3.49074815902218048554112600374, −2.28697815360706736910329634593, −1.36722351135435809534342521408,
0.41050940949349008742264155803, 2.04105714449523361009413947797, 3.10951951748663715111733188713, 3.88924753975672535977080093170, 4.80006837482447059919472290406, 5.56335000718488388194637683117, 6.44052764864756826990720648596, 7.43472276345850312496770778993, 8.225152367696728434549722499847, 8.624412740850947147045690494889