| L(s) = 1 | + 3·3-s + 10·5-s + 9·9-s − 12·11-s − 30·13-s + 30·15-s − 34·17-s − 148·19-s + 152·23-s − 25·25-s + 27·27-s − 106·29-s − 304·31-s − 36·33-s − 114·37-s − 90·39-s − 202·41-s + 116·43-s + 90·45-s − 224·47-s − 102·51-s − 274·53-s − 120·55-s − 444·57-s + 660·59-s − 382·61-s − 300·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.328·11-s − 0.640·13-s + 0.516·15-s − 0.485·17-s − 1.78·19-s + 1.37·23-s − 1/5·25-s + 0.192·27-s − 0.678·29-s − 1.76·31-s − 0.189·33-s − 0.506·37-s − 0.369·39-s − 0.769·41-s + 0.411·43-s + 0.298·45-s − 0.695·47-s − 0.280·51-s − 0.710·53-s − 0.294·55-s − 1.03·57-s + 1.45·59-s − 0.801·61-s − 0.572·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 148 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 304 T + p^{3} T^{2} \) |
| 37 | \( 1 + 114 T + p^{3} T^{2} \) |
| 41 | \( 1 + 202 T + p^{3} T^{2} \) |
| 43 | \( 1 - 116 T + p^{3} T^{2} \) |
| 47 | \( 1 + 224 T + p^{3} T^{2} \) |
| 53 | \( 1 + 274 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 382 T + p^{3} T^{2} \) |
| 67 | \( 1 - 12 T + p^{3} T^{2} \) |
| 71 | \( 1 + 552 T + p^{3} T^{2} \) |
| 73 | \( 1 - 614 T + p^{3} T^{2} \) |
| 79 | \( 1 - 880 T + p^{3} T^{2} \) |
| 83 | \( 1 - 108 T + p^{3} T^{2} \) |
| 89 | \( 1 - 86 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1426 T + p^{3} T^{2} \) |
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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
−9.073266279615146970935860247353, −8.312651532827438779228725071011, −7.28398162709486931122866993217, −6.57437889655017598603755127107, −5.54990669344408343352249177348, −4.70673047133107645052408363365, −3.58422768565965399465715008861, −2.40791092053197990181737315754, −1.75915552023851286490490503235, 0,
1.75915552023851286490490503235, 2.40791092053197990181737315754, 3.58422768565965399465715008861, 4.70673047133107645052408363365, 5.54990669344408343352249177348, 6.57437889655017598603755127107, 7.28398162709486931122866993217, 8.312651532827438779228725071011, 9.073266279615146970935860247353
