L(s) = 1 | − 2.41·5-s − 7-s + 2.41·11-s + 2.82·13-s − 2.82·17-s − 3.82·19-s + 5.24·23-s + 0.828·25-s + 2.82·29-s − 4.65·31-s + 2.41·35-s + 0.171·37-s + 2.07·41-s − 4.82·43-s + 0.343·47-s + 49-s + 4·53-s − 5.82·55-s − 3.65·59-s − 11.3·61-s − 6.82·65-s + 4.48·67-s + 5.58·71-s − 3.17·73-s − 2.41·77-s + 4.82·79-s + 10.8·83-s + ⋯ |
L(s) = 1 | − 1.07·5-s − 0.377·7-s + 0.727·11-s + 0.784·13-s − 0.685·17-s − 0.878·19-s + 1.09·23-s + 0.165·25-s + 0.525·29-s − 0.836·31-s + 0.408·35-s + 0.0282·37-s + 0.323·41-s − 0.736·43-s + 0.0500·47-s + 0.142·49-s + 0.549·53-s − 0.785·55-s − 0.476·59-s − 1.44·61-s − 0.846·65-s + 0.547·67-s + 0.662·71-s − 0.371·73-s − 0.275·77-s + 0.543·79-s + 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.276106394\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276106394\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 2.41T + 5T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 - 0.171T + 37T^{2} \) |
| 41 | \( 1 - 2.07T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 4.48T + 67T^{2} \) |
| 71 | \( 1 - 5.58T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 1.24T + 89T^{2} \) |
| 97 | \( 1 + 9.65T + 97T^{2} \) |
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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
−8.088178303615233497020290640247, −7.35833796135743173611486685856, −6.63633932199553747540173608425, −6.17452252461083247387039738026, −5.08333890997777209408527068602, −4.25715347616796096079096892586, −3.74020885916209061605834357685, −2.97376736223391030365707243738, −1.78577032724612582119414193948, −0.59330392068188946780710114287,
0.59330392068188946780710114287, 1.78577032724612582119414193948, 2.97376736223391030365707243738, 3.74020885916209061605834357685, 4.25715347616796096079096892586, 5.08333890997777209408527068602, 6.17452252461083247387039738026, 6.63633932199553747540173608425, 7.35833796135743173611486685856, 8.088178303615233497020290640247