L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 3·9-s − 11-s + 2·13-s + 2·14-s + 16-s + 17-s − 3·18-s − 4·19-s − 22-s − 6·23-s + 2·26-s + 2·28-s − 4·29-s − 2·31-s + 32-s + 34-s − 3·36-s + 4·37-s − 4·38-s − 2·41-s + 4·43-s − 44-s − 6·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 9-s − 0.301·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.707·18-s − 0.917·19-s − 0.213·22-s − 1.25·23-s + 0.392·26-s + 0.377·28-s − 0.742·29-s − 0.359·31-s + 0.176·32-s + 0.171·34-s − 1/2·36-s + 0.657·37-s − 0.648·38-s − 0.312·41-s + 0.609·43-s − 0.150·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.39497763355852317591654408181, −6.47875177658946888955749362433, −5.81766453832917562631456687489, −5.45157537021660920597227408686, −4.50241152982995792576061518303, −3.95910532862353522986446982764, −3.07080759554992805174547514682, −2.28484741230127991044054741388, −1.49015199008304796688155138715, 0,
1.49015199008304796688155138715, 2.28484741230127991044054741388, 3.07080759554992805174547514682, 3.95910532862353522986446982764, 4.50241152982995792576061518303, 5.45157537021660920597227408686, 5.81766453832917562631456687489, 6.47875177658946888955749362433, 7.39497763355852317591654408181