| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 5·11-s + 6·13-s + 16-s − 4·17-s − 4·19-s − 2·20-s + 5·22-s − 4·23-s − 25-s + 6·26-s + 7·29-s + 3·31-s + 32-s − 4·34-s + 8·37-s − 4·38-s − 2·40-s − 6·41-s + 8·43-s + 5·44-s − 4·46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 1.50·11-s + 1.66·13-s + 1/4·16-s − 0.970·17-s − 0.917·19-s − 0.447·20-s + 1.06·22-s − 0.834·23-s − 1/5·25-s + 1.17·26-s + 1.29·29-s + 0.538·31-s + 0.176·32-s − 0.685·34-s + 1.31·37-s − 0.648·38-s − 0.316·40-s − 0.937·41-s + 1.21·43-s + 0.753·44-s − 0.589·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.772141299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.772141299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.565965331263702843473468186588, −8.277975694086935115732027512853, −7.15009412751460627865473071223, −6.31529523399755536908292886005, −6.08719208937107993378577189797, −4.56909531034087080736198422303, −4.07442545737016836494288122854, −3.53214440962610527012228724866, −2.22570766730814268783848656685, −0.987749963759077104315662198648,
0.987749963759077104315662198648, 2.22570766730814268783848656685, 3.53214440962610527012228724866, 4.07442545737016836494288122854, 4.56909531034087080736198422303, 6.08719208937107993378577189797, 6.31529523399755536908292886005, 7.15009412751460627865473071223, 8.277975694086935115732027512853, 8.565965331263702843473468186588
