L(s) = 1 | − 7-s + 11-s + 13-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 − 91-s − 97-s + ⋯ |
L(s) = 1 | − 7-s + 11-s + 13-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 − 91-s − 97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.735873224\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.735873224\) |
\(L(1)\) |
\(\approx\) |
\(1.147147441\) |
\(L(1)\) |
\(\approx\) |
\(1.147147441\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 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
−28.8171372264551480782559072012, −27.87067861474959756546819489167, −26.865525824800293993438651130606, −25.5516551162428526571291856495, −25.16186251807120983569119536429, −23.53776231051619203315229688863, −22.86160558554397493322603808606, −21.74096692365282341174026509101, −20.67989463691621018473514931391, −19.42750683122014713375791919297, −18.79687203256083384587201578980, −17.30380473861409066033331850621, −16.4277349190353886928091180271, −15.34307497206724817155788495461, −14.11645167037890256849191233206, −13.01005586759749042453215746597, −11.976997034497005137370407321631, −10.65728172453007578033307010466, −9.49673843088330743017599881599, −8.43388154894637863228827315, −6.823052292953949953432353696699, −5.96082263611866179835864982702, −4.18962253151872300800023814866, −2.996698971090393673389675147582, −1.03888458426569614280228827842,
1.03888458426569614280228827842, 2.996698971090393673389675147582, 4.18962253151872300800023814866, 5.96082263611866179835864982702, 6.823052292953949953432353696699, 8.43388154894637863228827315, 9.49673843088330743017599881599, 10.65728172453007578033307010466, 11.976997034497005137370407321631, 13.01005586759749042453215746597, 14.11645167037890256849191233206, 15.34307497206724817155788495461, 16.4277349190353886928091180271, 17.30380473861409066033331850621, 18.79687203256083384587201578980, 19.42750683122014713375791919297, 20.67989463691621018473514931391, 21.74096692365282341174026509101, 22.86160558554397493322603808606, 23.53776231051619203315229688863, 25.16186251807120983569119536429, 25.5516551162428526571291856495, 26.865525824800293993438651130606, 27.87067861474959756546819489167, 28.8171372264551480782559072012