| L(s) = 1 | − 2·3-s − 3·7-s + 9-s − 4·13-s + 2·17-s + 2·19-s + 6·21-s − 23-s + 4·27-s − 6·29-s − 5·31-s − 8·37-s + 8·39-s + 41-s + 8·43-s − 5·47-s + 2·49-s − 4·51-s + 4·53-s − 4·57-s − 14·59-s + 14·61-s − 3·63-s + 2·67-s + 2·69-s + 8·71-s − 5·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.13·7-s + 1/3·9-s − 1.10·13-s + 0.485·17-s + 0.458·19-s + 1.30·21-s − 0.208·23-s + 0.769·27-s − 1.11·29-s − 0.898·31-s − 1.31·37-s + 1.28·39-s + 0.156·41-s + 1.21·43-s − 0.729·47-s + 2/7·49-s − 0.560·51-s + 0.549·53-s − 0.529·57-s − 1.82·59-s + 1.79·61-s − 0.377·63-s + 0.244·67-s + 0.240·69-s + 0.949·71-s − 0.585·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.37047270958952, −12.98838648617761, −12.42154645136751, −12.23477422781048, −11.80088036884422, −11.16249695393478, −10.83609432111476, −10.27204589029434, −9.851728711799774, −9.393675690479639, −9.060845596144160, −8.279450675284151, −7.659275384038202, −7.221336123449627, −6.744000178782498, −6.352681366654449, −5.601373763340992, −5.461047893362236, −4.978613699431547, −4.177841420091492, −3.687231508628598, −3.061920727141487, −2.536324001064308, −1.756100232778530, −0.9732237645877537, 0, 0,
0.9732237645877537, 1.756100232778530, 2.536324001064308, 3.061920727141487, 3.687231508628598, 4.177841420091492, 4.978613699431547, 5.461047893362236, 5.601373763340992, 6.352681366654449, 6.744000178782498, 7.221336123449627, 7.659275384038202, 8.279450675284151, 9.060845596144160, 9.393675690479639, 9.851728711799774, 10.27204589029434, 10.83609432111476, 11.16249695393478, 11.80088036884422, 12.23477422781048, 12.42154645136751, 12.98838648617761, 13.37047270958952
