L(s) = 1 | − 3-s + 4·5-s − 4·7-s + 9-s + 2·11-s + 13-s − 4·15-s − 6·17-s − 4·19-s + 4·21-s + 4·23-s + 11·25-s − 27-s + 6·29-s + 8·31-s − 2·33-s − 16·35-s + 10·37-s − 39-s − 4·41-s + 4·43-s + 4·45-s − 6·47-s + 9·49-s + 6·51-s − 6·53-s + 8·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 1.03·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 0.834·23-s + 11/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.348·33-s − 2.70·35-s + 1.64·37-s − 0.160·39-s − 0.624·41-s + 0.609·43-s + 0.596·45-s − 0.875·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.771788425\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.771788425\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.209632797526139354099334044597, −8.412081168783159735694253984596, −6.79976403646101930660690744681, −6.38643960892890947570913822009, −6.24285194557670647480016843307, −5.09754766043068360145456374630, −4.25781876644228347131524734341, −2.95526170418117470667875421111, −2.18289937290554630887181547294, −0.876666400119120651748065595063,
0.876666400119120651748065595063, 2.18289937290554630887181547294, 2.95526170418117470667875421111, 4.25781876644228347131524734341, 5.09754766043068360145456374630, 6.24285194557670647480016843307, 6.38643960892890947570913822009, 6.79976403646101930660690744681, 8.412081168783159735694253984596, 9.209632797526139354099334044597