| 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 + 24-s + 25-s + 26-s + 27-s − 28-s + 29-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 + 24-s + 25-s + 26-s + 27-s − 28-s + 29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.455362513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455362513\) |
| \(L(1)\) |
\(\approx\) |
\(1.965202054\) |
| \(L(1)\) |
\(\approx\) |
\(1.965202054\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 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
−38.68724240329872765497576852621, −37.96791783033085224078040530246, −36.07573583369179760813970017175, −34.87017245956883141460396113366, −33.15130794249205310083365342979, −31.91833869373659697245162297098, −31.25469395122993651116980291877, −30.10171925835832877824003242347, −28.5028654987024245497552475716, −26.48639123783683881594192774017, −25.462561010482497667429215273674, −23.94644729526781844527668854668, −22.87407602346311971158763595981, −21.20360225639625471304374327961, −19.98776684788736769840436487917, −18.98593120269587444083248143723, −15.93810136734798316256870844575, −15.38556073359858098799258622111, −13.60413172473855533286025588645, −12.581696786969624525139642734401, −10.63387123021858563292939521079, −8.334849030124398104335730457447, −6.731189150719542313772896829694, −4.21518980422971907980891242449, −2.871339848930367916555243647734,
2.871339848930367916555243647734, 4.21518980422971907980891242449, 6.731189150719542313772896829694, 8.334849030124398104335730457447, 10.63387123021858563292939521079, 12.581696786969624525139642734401, 13.60413172473855533286025588645, 15.38556073359858098799258622111, 15.93810136734798316256870844575, 18.98593120269587444083248143723, 19.98776684788736769840436487917, 21.20360225639625471304374327961, 22.87407602346311971158763595981, 23.94644729526781844527668854668, 25.462561010482497667429215273674, 26.48639123783683881594192774017, 28.5028654987024245497552475716, 30.10171925835832877824003242347, 31.25469395122993651116980291877, 31.91833869373659697245162297098, 33.15130794249205310083365342979, 34.87017245956883141460396113366, 36.07573583369179760813970017175, 37.96791783033085224078040530246, 38.68724240329872765497576852621
