L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.292i)11-s + 1.00i·14-s − 1.00·16-s + (0.292 + 0.707i)22-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)28-s + (−0.707 + 0.292i)29-s + (−0.707 − 0.707i)32-s + (−0.707 + 1.70i)37-s + (−1.70 − 0.707i)43-s + (−0.292 + 0.707i)44-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.292i)11-s + 1.00i·14-s − 1.00·16-s + (0.292 + 0.707i)22-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)28-s + (−0.707 + 0.292i)29-s + (−0.707 − 0.707i)32-s + (−0.707 + 1.70i)37-s + (−1.70 − 0.707i)43-s + (−0.292 + 0.707i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.792214215\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.792214215\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.240857230605533890157547885781, −8.584981061252636524316655868808, −8.034187419830536117085261529197, −6.99460249377852695177176683309, −6.46465848585577000637703961364, −5.47330184503695290307771097755, −4.84851355508664444668487386147, −4.01394670062281366855847775225, −2.95025200153962225945856468401, −1.86729051005627940736126416218,
1.15600757047971530270336309326, 2.11421982787502270904168949042, 3.54647635587387229175586828304, 3.93632007076975203731266967115, 5.08412844704054861597024899661, 5.61322515139075385520415627401, 6.75967460174509450190532711959, 7.36975742096602303812854120232, 8.469069308929077534750684460364, 9.331781664377649269267356736540