| L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s + 4·13-s + 16-s + 6·17-s + 3·20-s + 6·23-s + 4·25-s + 4·26-s − 8·29-s + 2·31-s + 32-s + 6·34-s − 9·37-s + 3·40-s − 4·41-s − 8·43-s + 6·46-s + 7·47-s − 7·49-s + 4·50-s + 4·52-s + 6·53-s − 8·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.670·20-s + 1.25·23-s + 4/5·25-s + 0.784·26-s − 1.48·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s − 1.47·37-s + 0.474·40-s − 0.624·41-s − 1.21·43-s + 0.884·46-s + 1.02·47-s − 49-s + 0.565·50-s + 0.554·52-s + 0.824·53-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.482674474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.482674474\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.552260510242655338973831592053, −7.52409818507437810651187235661, −6.80421601110623545995231790269, −6.01740605503031013045892043372, −5.51830083672446414400873481332, −4.93055375705784790328754095974, −3.67240277776344559521166335760, −3.12590129853361235274038077648, −1.95871910891761308621870242311, −1.23527885021980608506519637328,
1.23527885021980608506519637328, 1.95871910891761308621870242311, 3.12590129853361235274038077648, 3.67240277776344559521166335760, 4.93055375705784790328754095974, 5.51830083672446414400873481332, 6.01740605503031013045892043372, 6.80421601110623545995231790269, 7.52409818507437810651187235661, 8.552260510242655338973831592053
