| L(s) = 1 | − 4·3-s − 8·5-s − 14·7-s − 11·9-s − 8·11-s + 46·13-s + 32·15-s − 17·17-s + 116·19-s + 56·21-s + 94·23-s − 61·25-s + 152·27-s + 112·29-s − 50·31-s + 32·33-s + 112·35-s + 20·37-s − 184·39-s + 62·41-s + 68·43-s + 88·45-s + 60·47-s − 147·49-s + 68·51-s − 162·53-s + 64·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.715·5-s − 0.755·7-s − 0.407·9-s − 0.219·11-s + 0.981·13-s + 0.550·15-s − 0.242·17-s + 1.40·19-s + 0.581·21-s + 0.852·23-s − 0.487·25-s + 1.08·27-s + 0.717·29-s − 0.289·31-s + 0.168·33-s + 0.540·35-s + 0.0888·37-s − 0.755·39-s + 0.236·41-s + 0.241·43-s + 0.291·45-s + 0.186·47-s − 3/7·49-s + 0.186·51-s − 0.419·53-s + 0.156·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 5 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 94 T + p^{3} T^{2} \) |
| 29 | \( 1 - 112 T + p^{3} T^{2} \) |
| 31 | \( 1 + 50 T + p^{3} T^{2} \) |
| 37 | \( 1 - 20 T + p^{3} T^{2} \) |
| 41 | \( 1 - 62 T + p^{3} T^{2} \) |
| 43 | \( 1 - 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 60 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 724 T + p^{3} T^{2} \) |
| 61 | \( 1 - 388 T + p^{3} T^{2} \) |
| 67 | \( 1 - 172 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1090 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1062 T + p^{3} T^{2} \) |
| 79 | \( 1 + 114 T + p^{3} T^{2} \) |
| 83 | \( 1 + 68 T + p^{3} T^{2} \) |
| 89 | \( 1 + 666 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1322 T + p^{3} T^{2} \) |
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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
−9.086035324680272865956353843816, −8.239894670975427142720686382958, −7.33547096674975360677703527482, −6.44623868918390685292046726445, −5.72817224328968567862552990033, −4.82669673885679947053811932461, −3.65498622824339958543188919533, −2.88368850916098847849595200835, −1.05545098540561347773966479875, 0,
1.05545098540561347773966479875, 2.88368850916098847849595200835, 3.65498622824339958543188919533, 4.82669673885679947053811932461, 5.72817224328968567862552990033, 6.44623868918390685292046726445, 7.33547096674975360677703527482, 8.239894670975427142720686382958, 9.086035324680272865956353843816
