| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 21-s + 22-s − 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 21-s + 22-s − 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.146130765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.146130765\) |
| \(L(1)\) |
\(\approx\) |
\(1.393856345\) |
| \(L(1)\) |
\(\approx\) |
\(1.393856345\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 127 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 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
−28.52067572488685652646389057870, −27.99766520435093351585090067655, −26.59824644021116548642773324817, −25.30124695561747839302704554022, −24.19398541952146627356626474547, −23.279148337480309795448459607619, −22.677405948881897126576468287268, −21.96833579705921697263025438098, −20.58362189655130578482929961606, −19.539199884866128204208285284977, −18.52993437974209509579716166030, −16.74877019286484271434125709973, −16.161385682348337101787818022973, −15.33615762307803917923153995071, −13.89394810720922335386545359492, −12.631398354222797416029961828042, −11.90503815434017299792619603099, −11.10151564764325224425028448620, −9.76282214075559466375546575576, −7.70462897529502961342813667591, −6.581437893004037305522758931557, −5.72897321866236488034284711816, −4.19462861152323220660614436108, −3.40793170681041696251990926794, −1.035608606664081358328231063178,
1.035608606664081358328231063178, 3.40793170681041696251990926794, 4.19462861152323220660614436108, 5.72897321866236488034284711816, 6.581437893004037305522758931557, 7.70462897529502961342813667591, 9.76282214075559466375546575576, 11.10151564764325224425028448620, 11.90503815434017299792619603099, 12.631398354222797416029961828042, 13.89394810720922335386545359492, 15.33615762307803917923153995071, 16.161385682348337101787818022973, 16.74877019286484271434125709973, 18.52993437974209509579716166030, 19.539199884866128204208285284977, 20.58362189655130578482929961606, 21.96833579705921697263025438098, 22.677405948881897126576468287268, 23.279148337480309795448459607619, 24.19398541952146627356626474547, 25.30124695561747839302704554022, 26.59824644021116548642773324817, 27.99766520435093351585090067655, 28.52067572488685652646389057870
