L(s) = 1 | + 7-s − 11-s + 17-s + 19-s + 23-s + 29-s − 31-s − 37-s + 41-s + 43-s − 47-s + 49-s − 53-s − 59-s − 61-s − 67-s + 71-s + 73-s − 77-s + 79-s + 83-s + 89-s + 97-s + ⋯ |
L(s) = 1 | + 7-s − 11-s + 17-s + 19-s + 23-s + 29-s − 31-s − 37-s + 41-s + 43-s − 47-s + 49-s − 53-s − 59-s − 61-s − 67-s + 71-s + 73-s − 77-s + 79-s + 83-s + 89-s + 97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.678586127\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.678586127\) |
\(L(1)\) |
\(\approx\) |
\(1.272645822\) |
\(L(1)\) |
\(\approx\) |
\(1.272645822\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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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
−20.56494222053179448885057869694, −19.59284936695937273811928474682, −18.742801984884157368814261243069, −18.09464453606712872558899500501, −17.50630610704424279718105894641, −16.58346721117840886091580314412, −15.849655682680298359751258933312, −15.08046806491675659149204126875, −14.27304979337813339210666427266, −13.720656860950465307743859403270, −12.67619570062849114179071037410, −12.06253107453793570366081579172, −11.06602025964258717290869495543, −10.60030168800036189359233384306, −9.59873748338549394108967876124, −8.77067197774125362291393296653, −7.723523662352727843363013374882, −7.52591712706016223144439496153, −6.20540153991903214303817441961, −5.18208357224033577072698876383, −4.873434727256469188523245721137, −3.5449664720224192613210780956, −2.733739807912684513598254555156, −1.63585249527135430180419793473, −0.7275425887162805901150346916,
0.7275425887162805901150346916, 1.63585249527135430180419793473, 2.733739807912684513598254555156, 3.5449664720224192613210780956, 4.873434727256469188523245721137, 5.18208357224033577072698876383, 6.20540153991903214303817441961, 7.52591712706016223144439496153, 7.723523662352727843363013374882, 8.77067197774125362291393296653, 9.59873748338549394108967876124, 10.60030168800036189359233384306, 11.06602025964258717290869495543, 12.06253107453793570366081579172, 12.67619570062849114179071037410, 13.720656860950465307743859403270, 14.27304979337813339210666427266, 15.08046806491675659149204126875, 15.849655682680298359751258933312, 16.58346721117840886091580314412, 17.50630610704424279718105894641, 18.09464453606712872558899500501, 18.742801984884157368814261243069, 19.59284936695937273811928474682, 20.56494222053179448885057869694