| L(s) = 1 | + 3-s + 5-s + 9-s − 11-s + 2·13-s + 15-s + 17-s − 6·19-s + 6·23-s − 4·25-s + 27-s − 3·29-s − 5·31-s − 33-s + 4·37-s + 2·39-s + 6·41-s + 6·43-s + 45-s − 2·47-s + 51-s − 5·53-s − 55-s − 6·57-s + 3·59-s − 12·61-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s + 0.242·17-s − 1.37·19-s + 1.25·23-s − 4/5·25-s + 0.192·27-s − 0.557·29-s − 0.898·31-s − 0.174·33-s + 0.657·37-s + 0.320·39-s + 0.937·41-s + 0.914·43-s + 0.149·45-s − 0.291·47-s + 0.140·51-s − 0.686·53-s − 0.134·55-s − 0.794·57-s + 0.390·59-s − 1.53·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−13.39500054135577, −13.09202301756067, −12.67827183252736, −12.35511492228396, −11.42455535911319, −11.06906994360347, −10.73733643771908, −10.16559096095641, −9.529180086823895, −9.223303210354550, −8.830126432850014, −8.107027038623082, −7.847710290139597, −7.268480491969852, −6.589251181709176, −6.257056831911406, −5.597114234574014, −5.158668571889247, −4.457977318364884, −3.881964641704859, −3.499504599307327, −2.632623329639669, −2.315615736720156, −1.617578562311274, −0.9594003682917409, 0,
0.9594003682917409, 1.617578562311274, 2.315615736720156, 2.632623329639669, 3.499504599307327, 3.881964641704859, 4.457977318364884, 5.158668571889247, 5.597114234574014, 6.257056831911406, 6.589251181709176, 7.268480491969852, 7.847710290139597, 8.107027038623082, 8.830126432850014, 9.223303210354550, 9.529180086823895, 10.16559096095641, 10.73733643771908, 11.06906994360347, 11.42455535911319, 12.35511492228396, 12.67827183252736, 13.09202301756067, 13.39500054135577
