| L(s) = 1 | − 5·5-s + 6·7-s − 47·11-s − 5·13-s + 131·17-s + 56·19-s + 3·23-s + 25·25-s + 157·29-s − 225·31-s − 30·35-s − 70·37-s − 140·41-s − 397·43-s − 347·47-s − 307·49-s − 4·53-s + 235·55-s + 748·59-s − 338·61-s + 25·65-s − 492·67-s + 32·71-s + 970·73-s − 282·77-s + 1.25e3·79-s − 102·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.323·7-s − 1.28·11-s − 0.106·13-s + 1.86·17-s + 0.676·19-s + 0.0271·23-s + 1/5·25-s + 1.00·29-s − 1.30·31-s − 0.144·35-s − 0.311·37-s − 0.533·41-s − 1.40·43-s − 1.07·47-s − 0.895·49-s − 0.0103·53-s + 0.576·55-s + 1.65·59-s − 0.709·61-s + 0.0477·65-s − 0.897·67-s + 0.0534·71-s + 1.55·73-s − 0.417·77-s + 1.79·79-s − 0.134·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.736751650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736751650\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 47 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 T + p^{3} T^{2} \) |
| 17 | \( 1 - 131 T + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 23 | \( 1 - 3 T + p^{3} T^{2} \) |
| 29 | \( 1 - 157 T + p^{3} T^{2} \) |
| 31 | \( 1 + 225 T + p^{3} T^{2} \) |
| 37 | \( 1 + 70 T + p^{3} T^{2} \) |
| 41 | \( 1 + 140 T + p^{3} T^{2} \) |
| 43 | \( 1 + 397 T + p^{3} T^{2} \) |
| 47 | \( 1 + 347 T + p^{3} T^{2} \) |
| 53 | \( 1 + 4 T + p^{3} T^{2} \) |
| 59 | \( 1 - 748 T + p^{3} T^{2} \) |
| 61 | \( 1 + 338 T + p^{3} T^{2} \) |
| 67 | \( 1 + 492 T + p^{3} T^{2} \) |
| 71 | \( 1 - 32 T + p^{3} T^{2} \) |
| 73 | \( 1 - 970 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1257 T + p^{3} T^{2} \) |
| 83 | \( 1 + 102 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1488 T + p^{3} T^{2} \) |
| 97 | \( 1 - 974 T + p^{3} T^{2} \) |
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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
−8.498100785907236993859845443436, −7.890196878816479062806280722872, −7.42669317580197836844514959844, −6.39599039214083899139911304265, −5.16037531041611192514165527988, −5.10589647059892854649800396385, −3.60775903425659351766426725215, −3.03166797054684620898646913165, −1.77048020943707349065105844139, −0.59887782828199126544780922026,
0.59887782828199126544780922026, 1.77048020943707349065105844139, 3.03166797054684620898646913165, 3.60775903425659351766426725215, 5.10589647059892854649800396385, 5.16037531041611192514165527988, 6.39599039214083899139911304265, 7.42669317580197836844514959844, 7.890196878816479062806280722872, 8.498100785907236993859845443436
