L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 11-s + 12-s + 13-s − 14-s + 16-s − 6·17-s − 18-s − 4·19-s + 21-s − 22-s − 24-s − 5·25-s − 26-s + 27-s + 28-s + 6·29-s + 2·31-s − 32-s + 33-s + 6·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.213·22-s − 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.939600956213806596881204281900, −7.04576517766578426769797044952, −6.56890271020717317917775066153, −5.76664171057413663377072465217, −4.62921816402815364000416993243, −4.07250101882315061852419144352, −3.02454295717369388295349502844, −2.16443810714876659809131045446, −1.45233236678402023597389627869, 0,
1.45233236678402023597389627869, 2.16443810714876659809131045446, 3.02454295717369388295349502844, 4.07250101882315061852419144352, 4.62921816402815364000416993243, 5.76664171057413663377072465217, 6.56890271020717317917775066153, 7.04576517766578426769797044952, 7.939600956213806596881204281900