L(s) = 1 | − 0.517·2-s − 1.73·4-s + (−2.44 − i)7-s + 1.93·8-s + 0.378i·11-s + 1.79·13-s + (1.26 + 0.517i)14-s + 2.46·16-s − 3.46i·17-s − 1.79i·19-s − 0.196i·22-s + 1.41·23-s − 0.928·26-s + (4.24 + 1.73i)28-s + 1.41i·29-s + ⋯ |
L(s) = 1 | − 0.366·2-s − 0.866·4-s + (−0.925 − 0.377i)7-s + 0.683·8-s + 0.114i·11-s + 0.497·13-s + (0.338 + 0.138i)14-s + 0.616·16-s − 0.840i·17-s − 0.411i·19-s − 0.0418i·22-s + 0.294·23-s − 0.182·26-s + (0.801 + 0.327i)28-s + 0.262i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1266964716\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1266964716\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.44 + i)T \) |
good | 2 | \( 1 + 0.517T + 2T^{2} \) |
| 11 | \( 1 - 0.378iT - 11T^{2} \) |
| 13 | \( 1 - 1.79T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 1.79iT - 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 6.69iT - 31T^{2} \) |
| 37 | \( 1 - 1.46iT - 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4.92iT - 43T^{2} \) |
| 47 | \( 1 + 9.46iT - 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 13.3iT - 61T^{2} \) |
| 67 | \( 1 - 10.9iT - 67T^{2} \) |
| 71 | \( 1 - 15.0iT - 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 9.46iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−9.767439242312131603293777019094, −8.910455787969886721497107569064, −8.454588902522019823211853692091, −7.26844453306963147089030801070, −6.77439886409933919793196456575, −5.57501325470260864768793573767, −4.77797339401540827315533289805, −3.79993277671037013996895599753, −2.95883784351812994237204302556, −1.24848386927671274572367812387,
0.06460121332706287715788170124, 1.60613255278583907782751792913, 3.13971380450333657125841400919, 3.92402899628118943295083517862, 4.91735930015728027925878425555, 5.94909927234193054932542105986, 6.51904330909924118092987718836, 7.84889915918871780792288048675, 8.269148203199063952982901139391, 9.363259410810302723273007022445