L(s) = 1 | − 2-s + (0.5 − 0.866i)3-s + 4-s + 5-s + (−0.5 + 0.866i)6-s − 8-s + (−0.5 − 0.866i)9-s − 10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + 16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + ⋯ |
L(s) = 1 | − 2-s + (0.5 − 0.866i)3-s + 4-s + 5-s + (−0.5 + 0.866i)6-s − 8-s + (−0.5 − 0.866i)9-s − 10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + 16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7058855531 - 1.175309219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7058855531 - 1.175309219i\) |
\(L(1)\) |
\(\approx\) |
\(0.8330280454 - 0.4433451230i\) |
\(L(1)\) |
\(\approx\) |
\(0.8330280454 - 0.4433451230i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.53091195552334117677438486001, −27.68906749228959845430226431221, −26.58837012222332937545028215986, −25.77664699803654671776175965216, −25.32553777056727669034928963409, −24.037114052841645371779704446286, −22.39816967505542648196411159654, −21.241674149841273061500957148517, −20.701563281298558701243589812248, −19.713788033704352192193747909623, −18.41927441584943990742016655038, −17.554384912425515424715368190109, −16.45286783778003514969053888971, −15.6220345028947088121380457098, −14.47637994310207430419456721982, −13.33908901361964791429527384929, −11.60791264485177659065069348596, −10.451946095877102338375734165223, −9.60726716075645193750745581975, −8.937752003491870251745872737124, −7.565639325238854061155383297706, −6.18382883791823198260760570643, −4.731693384817567148884829712600, −2.87596615845942148419023058637, −1.773145505350909677619408070090,
0.68406036945237382488549564323, 2.01510492683497910985360365948, 3.08901546627084844258241391393, 5.83865949900599474804340592399, 6.55625555663482546324909189344, 8.082831180513851830069757146461, 8.67479067325327974637980474297, 10.01453594138933398649180024501, 11.01574505534965831667382879497, 12.52603313391776594428958656652, 13.448316524987463398497652338799, 14.62553258434143938933260906960, 15.907111144865688729961020461, 17.235566720417169226062668711831, 17.97876446699219570182621531358, 18.734028461599712767989923942088, 19.796963060801563260906123904689, 20.73444903089732916833687003524, 21.67313170485889273520050901893, 23.406798033167735076643186432414, 24.58084738433232237720209335173, 25.0523539765298843846434195719, 26.11735343958197831906028600024, 26.69607031233818128803126790792, 28.354069784808133843898490827226