| L(s) = 1 | + 5-s − 7-s − 11-s − 2·17-s − 6·19-s − 3·23-s + 25-s + 29-s − 5·31-s − 35-s + 2·37-s + 3·41-s − 11·43-s − 3·47-s − 6·49-s + 11·53-s − 55-s − 14·59-s − 14·61-s + 4·67-s + 10·71-s + 11·73-s + 77-s + 8·79-s + 16·83-s − 2·85-s + 2·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.301·11-s − 0.485·17-s − 1.37·19-s − 0.625·23-s + 1/5·25-s + 0.185·29-s − 0.898·31-s − 0.169·35-s + 0.328·37-s + 0.468·41-s − 1.67·43-s − 0.437·47-s − 6/7·49-s + 1.51·53-s − 0.134·55-s − 1.82·59-s − 1.79·61-s + 0.488·67-s + 1.18·71-s + 1.28·73-s + 0.113·77-s + 0.900·79-s + 1.75·83-s − 0.216·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.87044404297898, −12.35102479199930, −12.09901513745419, −11.27699372003723, −11.02479924202424, −10.50132043424776, −10.15952074493771, −9.586510878475787, −9.211111575731203, −8.753165403792900, −8.197554913731382, −7.840945845820708, −7.237951079526427, −6.641198640997429, −6.204169876010116, −6.077689082158010, −5.112856932972985, −4.940195965312316, −4.265931512608832, −3.685620613429816, −3.242002914288632, −2.507360735143927, −2.036238251994992, −1.623088722649558, −0.6285913839530939, 0,
0.6285913839530939, 1.623088722649558, 2.036238251994992, 2.507360735143927, 3.242002914288632, 3.685620613429816, 4.265931512608832, 4.940195965312316, 5.112856932972985, 6.077689082158010, 6.204169876010116, 6.641198640997429, 7.237951079526427, 7.840945845820708, 8.197554913731382, 8.753165403792900, 9.211111575731203, 9.586510878475787, 10.15952074493771, 10.50132043424776, 11.02479924202424, 11.27699372003723, 12.09901513745419, 12.35102479199930, 12.87044404297898
