L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·11-s − 5·13-s + 14-s + 16-s + 3·17-s + 2·19-s + 3·22-s − 3·23-s + 5·26-s − 28-s + 9·29-s + 5·31-s − 32-s − 3·34-s − 2·37-s − 2·38-s + 43-s − 3·44-s + 3·46-s − 3·47-s + 49-s − 5·52-s − 6·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.904·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s + 0.639·22-s − 0.625·23-s + 0.980·26-s − 0.188·28-s + 1.67·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s − 0.328·37-s − 0.324·38-s + 0.152·43-s − 0.452·44-s + 0.442·46-s − 0.437·47-s + 1/7·49-s − 0.693·52-s − 0.824·53-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.65618445671278159714331294675, −6.70569711545962756978311191138, −6.22422863765101683167858326102, −5.18621025301357460716122900980, −4.83315468504571157321888742699, −3.62494527571383278725762814448, −2.80120423549663111481794655299, −2.27731908344793154553143725477, −1.03188740768723227668891415608, 0,
1.03188740768723227668891415608, 2.27731908344793154553143725477, 2.80120423549663111481794655299, 3.62494527571383278725762814448, 4.83315468504571157321888742699, 5.18621025301357460716122900980, 6.22422863765101683167858326102, 6.70569711545962756978311191138, 7.65618445671278159714331294675