L(s) = 1 | + 3-s − 1.04·5-s + 9-s + 5.93·11-s − 0.888·13-s − 1.04·15-s + 4.51·17-s + 3.47·19-s − 1.93·23-s − 3.91·25-s + 27-s − 8.38·29-s + 5.13·31-s + 5.93·33-s + 5.65·37-s − 0.888·39-s + 8.39·41-s − 0.743·43-s − 1.04·45-s − 4.21·47-s + 4.51·51-s − 6.91·53-s − 6.18·55-s + 3.47·57-s + 5.43·59-s + 10.9·61-s + 0.926·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.466·5-s + 0.333·9-s + 1.78·11-s − 0.246·13-s − 0.269·15-s + 1.09·17-s + 0.797·19-s − 0.402·23-s − 0.782·25-s + 0.192·27-s − 1.55·29-s + 0.921·31-s + 1.03·33-s + 0.929·37-s − 0.142·39-s + 1.31·41-s − 0.113·43-s − 0.155·45-s − 0.615·47-s + 0.632·51-s − 0.949·53-s − 0.833·55-s + 0.460·57-s + 0.708·59-s + 1.40·61-s + 0.114·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.623848425\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.623848425\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.04T + 5T^{2} \) |
| 11 | \( 1 - 5.93T + 11T^{2} \) |
| 13 | \( 1 + 0.888T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 23 | \( 1 + 1.93T + 23T^{2} \) |
| 29 | \( 1 + 8.38T + 29T^{2} \) |
| 31 | \( 1 - 5.13T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 + 0.743T + 43T^{2} \) |
| 47 | \( 1 + 4.21T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 - 5.43T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 6.20T + 67T^{2} \) |
| 71 | \( 1 - 3.72T + 71T^{2} \) |
| 73 | \( 1 + 3.49T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 + 5.00T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.106956784570376887940048160344, −7.73768623980308319543222665619, −6.96868209042394186729496883891, −6.18091004917762338866795233271, −5.40878211144539251603515444114, −4.26153808730787182486091464861, −3.81242805297552321738360144825, −3.04627472666482723141256637447, −1.86719160822634474275619884883, −0.924349618637425819444354498474,
0.924349618637425819444354498474, 1.86719160822634474275619884883, 3.04627472666482723141256637447, 3.81242805297552321738360144825, 4.26153808730787182486091464861, 5.40878211144539251603515444114, 6.18091004917762338866795233271, 6.96868209042394186729496883891, 7.73768623980308319543222665619, 8.106956784570376887940048160344