| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 4·11-s − 2·13-s + 16-s − 17-s + 4·19-s + 2·20-s − 4·22-s − 25-s + 2·26-s + 10·29-s + 8·31-s − 32-s + 34-s − 2·37-s − 4·38-s − 2·40-s − 10·41-s + 12·43-s + 4·44-s − 7·49-s + 50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s + 1.85·29-s + 1.43·31-s − 0.176·32-s + 0.171·34-s − 0.328·37-s − 0.648·38-s − 0.316·40-s − 1.56·41-s + 1.82·43-s + 0.603·44-s − 49-s + 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.147174036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147174036\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−11.73531253806048906399243423876, −10.52566566553203306448675052964, −9.723242141956922614700830245165, −9.104444876573252449353403617805, −8.000937863938982294341146593643, −6.79533660929922109224513153399, −6.05886413503827459123865903257, −4.66119827871442481374467622682, −2.92323665285260202872977027719, −1.43438626728056627432143840048,
1.43438626728056627432143840048, 2.92323665285260202872977027719, 4.66119827871442481374467622682, 6.05886413503827459123865903257, 6.79533660929922109224513153399, 8.000937863938982294341146593643, 9.104444876573252449353403617805, 9.723242141956922614700830245165, 10.52566566553203306448675052964, 11.73531253806048906399243423876
