| L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s − 2·7-s + 8-s + 9-s − 4·10-s + 11-s + 12-s + 4·13-s − 2·14-s − 4·15-s + 16-s − 2·17-s + 18-s − 4·20-s − 2·21-s + 22-s − 6·23-s + 24-s + 11·25-s + 4·26-s + 27-s − 2·28-s + 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.534·14-s − 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.894·20-s − 0.436·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.784·26-s + 0.192·27-s − 0.377·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.191614815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.191614815\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.93063692211965970387372568841, −13.79927171797322583714953112703, −12.63981185671263095706904211379, −11.77772233494394727351761316190, −10.63196865714989964750883164862, −8.822664775845247131815531688861, −7.72339303053252541131209387915, −6.46917181705411800265373292912, −4.25842178547984493307347569445, −3.34833436263269527102959669443,
3.34833436263269527102959669443, 4.25842178547984493307347569445, 6.46917181705411800265373292912, 7.72339303053252541131209387915, 8.822664775845247131815531688861, 10.63196865714989964750883164862, 11.77772233494394727351761316190, 12.63981185671263095706904211379, 13.79927171797322583714953112703, 14.93063692211965970387372568841
