| L(s) = 1 | − 2-s − 1.02·3-s + 4-s − 1.91·5-s + 1.02·6-s + 0.691·7-s − 8-s − 1.94·9-s + 1.91·10-s − 11-s − 1.02·12-s − 0.362·13-s − 0.691·14-s + 1.95·15-s + 16-s − 6.99·17-s + 1.94·18-s + 2.28·19-s − 1.91·20-s − 0.709·21-s + 22-s + 6.46·23-s + 1.02·24-s − 1.35·25-s + 0.362·26-s + 5.07·27-s + 0.691·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.591·3-s + 0.5·4-s − 0.854·5-s + 0.418·6-s + 0.261·7-s − 0.353·8-s − 0.649·9-s + 0.604·10-s − 0.301·11-s − 0.295·12-s − 0.100·13-s − 0.184·14-s + 0.505·15-s + 0.250·16-s − 1.69·17-s + 0.459·18-s + 0.523·19-s − 0.427·20-s − 0.154·21-s + 0.213·22-s + 1.34·23-s + 0.209·24-s − 0.270·25-s + 0.0711·26-s + 0.976·27-s + 0.130·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3619394320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3619394320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 1.02T + 3T^{2} \) |
| 5 | \( 1 + 1.91T + 5T^{2} \) |
| 7 | \( 1 - 0.691T + 7T^{2} \) |
| 13 | \( 1 + 0.362T + 13T^{2} \) |
| 17 | \( 1 + 6.99T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 - 6.46T + 23T^{2} \) |
| 29 | \( 1 - 1.00T + 29T^{2} \) |
| 31 | \( 1 + 8.75T + 31T^{2} \) |
| 37 | \( 1 + 2.02T + 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 + 5.96T + 43T^{2} \) |
| 47 | \( 1 + 4.81T + 47T^{2} \) |
| 53 | \( 1 + 6.54T + 53T^{2} \) |
| 59 | \( 1 - 3.27T + 59T^{2} \) |
| 61 | \( 1 - 4.54T + 61T^{2} \) |
| 67 | \( 1 - 1.96T + 67T^{2} \) |
| 71 | \( 1 - 3.77T + 71T^{2} \) |
| 73 | \( 1 + 6.91T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 6.39T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 - 3.87T + 97T^{2} \) |
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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
−8.392859763049711909557456821326, −7.73826541236266453076675499460, −6.92763043108177499705997907652, −6.44389475670909242733170803911, −5.32188684676203366753412061533, −4.85364063354941796393232246909, −3.71591038262496974633720159350, −2.86077916790723144879800330637, −1.76947861882022580110865711447, −0.37617183433079543697671581158,
0.37617183433079543697671581158, 1.76947861882022580110865711447, 2.86077916790723144879800330637, 3.71591038262496974633720159350, 4.85364063354941796393232246909, 5.32188684676203366753412061533, 6.44389475670909242733170803911, 6.92763043108177499705997907652, 7.73826541236266453076675499460, 8.392859763049711909557456821326
