L(s) = 1 | + 5-s + 7-s + 11-s − 13-s − 17-s − 19-s − 23-s + 25-s + 29-s + 31-s + 35-s − 37-s − 41-s − 43-s − 47-s + 49-s + 53-s + 55-s + 59-s − 61-s − 65-s − 67-s − 71-s + 73-s + 77-s + 79-s + 83-s + ⋯ |
L(s) = 1 | + 5-s + 7-s + 11-s − 13-s − 17-s − 19-s − 23-s + 25-s + 29-s + 31-s + 35-s − 37-s − 41-s − 43-s − 47-s + 49-s + 53-s + 55-s + 59-s − 61-s − 65-s − 67-s − 71-s + 73-s + 77-s + 79-s + 83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.571915159\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.571915159\) |
\(L(1)\) |
\(\approx\) |
\(1.282549830\) |
\(L(1)\) |
\(\approx\) |
\(1.282549830\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 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
−38.14247751933542285576614257138, −37.169059080332133559649624087507, −36.06433119967803861641008437474, −34.34563209016533977564689216556, −33.37149707342212421697214612912, −32.11095898713639789271857905798, −30.4826325439914416234779210549, −29.47042426859464920571783262335, −27.99747354563209215912756582706, −26.69287992206634801816628273213, −25.114013937714922427003144983077, −24.18502420176868196704264889019, −22.24400758994810938440801208316, −21.23481315500759825470276937857, −19.7240153244415519968464947702, −17.87462416651816148400063708409, −17.030405571434057892804535988390, −14.88654143969881284582619693881, −13.75222966580168075032909065230, −11.935859369599332562538818607605, −10.231110375567980986681669522485, −8.631859564899361390761044162516, −6.55891715947090437049580839541, −4.72270950581234428327684375536, −1.97719051443795299435619611950,
1.97719051443795299435619611950, 4.72270950581234428327684375536, 6.55891715947090437049580839541, 8.631859564899361390761044162516, 10.231110375567980986681669522485, 11.935859369599332562538818607605, 13.75222966580168075032909065230, 14.88654143969881284582619693881, 17.030405571434057892804535988390, 17.87462416651816148400063708409, 19.7240153244415519968464947702, 21.23481315500759825470276937857, 22.24400758994810938440801208316, 24.18502420176868196704264889019, 25.114013937714922427003144983077, 26.69287992206634801816628273213, 27.99747354563209215912756582706, 29.47042426859464920571783262335, 30.4826325439914416234779210549, 32.11095898713639789271857905798, 33.37149707342212421697214612912, 34.34563209016533977564689216556, 36.06433119967803861641008437474, 37.169059080332133559649624087507, 38.14247751933542285576614257138