| L(s) = 1 | − 3-s − 3.58·5-s − 3.12·7-s + 9-s + 0.0194·11-s − 5.12·13-s + 3.58·15-s − 1.02·17-s + 7.39·19-s + 3.12·21-s − 8.31·23-s + 7.83·25-s − 27-s + 9.35·29-s + 1.12·31-s − 0.0194·33-s + 11.2·35-s − 0.220·37-s + 5.12·39-s + 4.18·41-s − 3.62·43-s − 3.58·45-s + 8.06·47-s + 2.78·49-s + 1.02·51-s + 5.02·53-s − 0.0698·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.60·5-s − 1.18·7-s + 0.333·9-s + 0.00587·11-s − 1.42·13-s + 0.925·15-s − 0.249·17-s + 1.69·19-s + 0.682·21-s − 1.73·23-s + 1.56·25-s − 0.192·27-s + 1.73·29-s + 0.201·31-s − 0.00339·33-s + 1.89·35-s − 0.0362·37-s + 0.821·39-s + 0.653·41-s − 0.552·43-s − 0.534·45-s + 1.17·47-s + 0.397·49-s + 0.144·51-s + 0.690·53-s − 0.00941·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 5 | \( 1 + 3.58T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 0.0194T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 + 1.02T + 17T^{2} \) |
| 19 | \( 1 - 7.39T + 19T^{2} \) |
| 23 | \( 1 + 8.31T + 23T^{2} \) |
| 29 | \( 1 - 9.35T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 + 0.220T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 + 3.62T + 43T^{2} \) |
| 47 | \( 1 - 8.06T + 47T^{2} \) |
| 53 | \( 1 - 5.02T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 8.59T + 61T^{2} \) |
| 67 | \( 1 - 3.59T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 4.86T + 73T^{2} \) |
| 79 | \( 1 - 6.22T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69038296968450271016904889694, −6.87196007334873567547306400082, −6.35021546218145912812010674556, −5.37987129701571621841582108698, −4.66001042598534817250095060755, −3.97296026247839777148470194000, −3.25834614483719618631236688404, −2.48260542231300582468667897296, −0.813920274229178494924943372321, 0,
0.813920274229178494924943372321, 2.48260542231300582468667897296, 3.25834614483719618631236688404, 3.97296026247839777148470194000, 4.66001042598534817250095060755, 5.37987129701571621841582108698, 6.35021546218145912812010674556, 6.87196007334873567547306400082, 7.69038296968450271016904889694
