L(s) = 1 | + (0.564 − 0.825i)3-s + (0.362 − 0.931i)5-s + (−0.985 + 0.170i)7-s + (−0.362 − 0.931i)9-s + (−0.466 + 0.884i)13-s + (−0.564 − 0.825i)15-s + (−0.0855 + 0.996i)17-s + (−0.921 + 0.389i)19-s + (−0.415 + 0.909i)21-s + (−0.736 − 0.676i)25-s + (−0.974 − 0.226i)27-s + (−0.921 − 0.389i)29-s + (−0.941 − 0.336i)31-s + (−0.198 + 0.980i)35-s + (0.254 − 0.967i)37-s + ⋯ |
L(s) = 1 | + (0.564 − 0.825i)3-s + (0.362 − 0.931i)5-s + (−0.985 + 0.170i)7-s + (−0.362 − 0.931i)9-s + (−0.466 + 0.884i)13-s + (−0.564 − 0.825i)15-s + (−0.0855 + 0.996i)17-s + (−0.921 + 0.389i)19-s + (−0.415 + 0.909i)21-s + (−0.736 − 0.676i)25-s + (−0.974 − 0.226i)27-s + (−0.921 − 0.389i)29-s + (−0.941 − 0.336i)31-s + (−0.198 + 0.980i)35-s + (0.254 − 0.967i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1097331261 - 0.2928917970i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1097331261 - 0.2928917970i\) |
\(L(1)\) |
\(\approx\) |
\(0.7920471999 - 0.3695665079i\) |
\(L(1)\) |
\(\approx\) |
\(0.7920471999 - 0.3695665079i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.564 - 0.825i)T \) |
| 5 | \( 1 + (0.362 - 0.931i)T \) |
| 7 | \( 1 + (-0.985 + 0.170i)T \) |
| 13 | \( 1 + (-0.466 + 0.884i)T \) |
| 17 | \( 1 + (-0.0855 + 0.996i)T \) |
| 19 | \( 1 + (-0.921 + 0.389i)T \) |
| 29 | \( 1 + (-0.921 - 0.389i)T \) |
| 31 | \( 1 + (-0.941 - 0.336i)T \) |
| 37 | \( 1 + (0.254 - 0.967i)T \) |
| 41 | \( 1 + (-0.254 - 0.967i)T \) |
| 43 | \( 1 + (-0.959 - 0.281i)T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.696 + 0.717i)T \) |
| 59 | \( 1 + (-0.897 + 0.441i)T \) |
| 61 | \( 1 + (-0.610 + 0.791i)T \) |
| 67 | \( 1 + (0.415 - 0.909i)T \) |
| 71 | \( 1 + (-0.198 - 0.980i)T \) |
| 73 | \( 1 + (0.516 - 0.856i)T \) |
| 79 | \( 1 + (-0.466 + 0.884i)T \) |
| 83 | \( 1 + (-0.998 + 0.0570i)T \) |
| 89 | \( 1 + (0.959 + 0.281i)T \) |
| 97 | \( 1 + (0.998 + 0.0570i)T \) |
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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
−21.90739129167703171643169440033, −21.65373747598450115847844719407, −20.29066206468753842220784822630, −20.016235207437743165400600074246, −19.00040184171850135310264534305, −18.3809441918914225314452546425, −17.279783009512638042957823272397, −16.54768089141007603942800841829, −15.67611818624562106946925626513, −14.997248090684541600679432836570, −14.398399351293125671851999483643, −13.40564512307727685861639304543, −12.893045333587911044234599416440, −11.46239935514027072682046489602, −10.71699843539796174235338960239, −9.917718278340171697460950497608, −9.52456903065340774697701114349, −8.448942362362163749210086240283, −7.390582590315175491172075277011, −6.64635245167062933557895586543, −5.598082267349938242922427820574, −4.67582327030176532212315376901, −3.39855998271005850837497892817, −3.038559076348365381922226094388, −2.055609255687157168576423575790,
0.10831322707551451518974287709, 1.659842921261762311989288403416, 2.20729407063329417753619832442, 3.52239208907168854800428066750, 4.34058770390855381773509748155, 5.76761529291036909758302571235, 6.28955087610093240687649669392, 7.28406453848776625641360636600, 8.20298290408882541995723182076, 9.13553800232512661350006306393, 9.426879023651865365840088778469, 10.6461281069105830767754085102, 11.95393915978059430122607920744, 12.59675599860879547900698392396, 13.050219155538592108922393982821, 13.85935234432164250819306420928, 14.738687507090944744067325361238, 15.576820207887265174811231246208, 16.81150183567167430598989466485, 16.950816721926138658621880192328, 18.15745595058681291587564592230, 19.00166392162848289548520204487, 19.52065899707920794291661156862, 20.20163095063813583878733700137, 21.11265894126369655447448144979