| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·11-s − 13-s + 14-s + 16-s − 17-s − 2·19-s + 3·22-s + 7·23-s + 26-s − 28-s + 5·29-s − 5·31-s − 32-s + 34-s + 2·37-s + 2·38-s − 6·41-s − 5·43-s − 3·44-s − 7·46-s + 47-s + 49-s − 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.904·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s + 0.639·22-s + 1.45·23-s + 0.196·26-s − 0.188·28-s + 0.928·29-s − 0.898·31-s − 0.176·32-s + 0.171·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s − 0.762·43-s − 0.452·44-s − 1.03·46-s + 0.145·47-s + 1/7·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9024754729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9024754729\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.76176211554912384006815497285, −6.97939308076693384933490400963, −6.64381915657485406465574189181, −5.64268527822678172032225215595, −5.08022280376576453253784406624, −4.20199196525465699912553674754, −3.14589664382766435283476935224, −2.62608844781030582939481587868, −1.65268083037588399601065474897, −0.49862046587507905904134075863,
0.49862046587507905904134075863, 1.65268083037588399601065474897, 2.62608844781030582939481587868, 3.14589664382766435283476935224, 4.20199196525465699912553674754, 5.08022280376576453253784406624, 5.64268527822678172032225215595, 6.64381915657485406465574189181, 6.97939308076693384933490400963, 7.76176211554912384006815497285
