L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s + 3·11-s − 5·13-s + 16-s − 2·17-s − 3·18-s − 5·19-s + 3·22-s − 7·23-s − 5·26-s − 4·29-s − 2·31-s + 32-s − 2·34-s − 3·36-s + 37-s − 5·38-s + 3·41-s + 2·43-s + 3·44-s − 7·46-s − 7·47-s − 5·52-s + 9·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s + 0.904·11-s − 1.38·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 1.14·19-s + 0.639·22-s − 1.45·23-s − 0.980·26-s − 0.742·29-s − 0.359·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s + 0.164·37-s − 0.811·38-s + 0.468·41-s + 0.304·43-s + 0.452·44-s − 1.03·46-s − 1.02·47-s − 0.693·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.529327395328125300035277377291, −7.71270410642732609357976710037, −6.85883528817855510764191066515, −6.13604350311255942308493216313, −5.45331374100094683523798989931, −4.46796068793129760508479264441, −3.82683913114427314553137035394, −2.66837975770199555076166787902, −1.93042364424011045388424200694, 0,
1.93042364424011045388424200694, 2.66837975770199555076166787902, 3.82683913114427314553137035394, 4.46796068793129760508479264441, 5.45331374100094683523798989931, 6.13604350311255942308493216313, 6.85883528817855510764191066515, 7.71270410642732609357976710037, 8.529327395328125300035277377291