L(s) = 1 | + 2-s + 4-s + 8-s − 11-s − 2·13-s + 16-s − 17-s − 4·19-s − 22-s − 23-s − 5·25-s − 2·26-s + 3·29-s − 10·31-s + 32-s − 34-s − 3·37-s − 4·38-s + 5·41-s − 6·43-s − 44-s − 46-s − 6·47-s − 7·49-s − 5·50-s − 2·52-s + 9·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.301·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.213·22-s − 0.208·23-s − 25-s − 0.392·26-s + 0.557·29-s − 1.79·31-s + 0.176·32-s − 0.171·34-s − 0.493·37-s − 0.648·38-s + 0.780·41-s − 0.914·43-s − 0.150·44-s − 0.147·46-s − 0.875·47-s − 49-s − 0.707·50-s − 0.277·52-s + 1.23·53-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 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
−7.996938913383203314870290154235, −7.22206638291354805981657247853, −6.57515783607279139845188294527, −5.74081923658863157752543849210, −5.09217295609891210161177252156, −4.25706198657269990215153911560, −3.54115789365286169577801042656, −2.50548656081883446637100970663, −1.74162271240657114429903897651, 0,
1.74162271240657114429903897651, 2.50548656081883446637100970663, 3.54115789365286169577801042656, 4.25706198657269990215153911560, 5.09217295609891210161177252156, 5.74081923658863157752543849210, 6.57515783607279139845188294527, 7.22206638291354805981657247853, 7.996938913383203314870290154235