| L(s) = 1 | − 0.221·3-s − 22.7·7-s − 26.9·9-s + 66.8·11-s + 32.9·13-s − 35.9·17-s − 44.0·19-s + 5.04·21-s − 139.·23-s + 11.9·27-s + 134.·29-s − 229.·31-s − 14.8·33-s − 79.1·37-s − 7.29·39-s + 384.·41-s + 251.·43-s + 241.·47-s + 176.·49-s + 7.96·51-s + 222·53-s + 9.75·57-s + 552.·59-s + 494.·61-s + 614.·63-s − 574.·67-s + 30.7·69-s + ⋯ |
| L(s) = 1 | − 0.0426·3-s − 1.23·7-s − 0.998·9-s + 1.83·11-s + 0.702·13-s − 0.512·17-s − 0.531·19-s + 0.0524·21-s − 1.26·23-s + 0.0851·27-s + 0.863·29-s − 1.32·31-s − 0.0780·33-s − 0.351·37-s − 0.0299·39-s + 1.46·41-s + 0.890·43-s + 0.749·47-s + 0.515·49-s + 0.0218·51-s + 0.575·53-s + 0.0226·57-s + 1.21·59-s + 1.03·61-s + 1.22·63-s − 1.04·67-s + 0.0537·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.494553069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494553069\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.221T + 27T^{2} \) |
| 7 | \( 1 + 22.7T + 343T^{2} \) |
| 11 | \( 1 - 66.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 79.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 384.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 251.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 241.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 222T + 1.48e5T^{2} \) |
| 59 | \( 1 - 552.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 494.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 574.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 654.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 179.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 810.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 29.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 383.T + 9.12e5T^{2} \) |
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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
−9.707726329741287678388099473700, −9.032837951737196726740129131865, −8.443038312056909725707071466074, −7.05773216835812314713426719636, −6.28306870048384612133156971537, −5.80434565975447127773988177363, −4.13717434369415984791877706804, −3.53143815977278326719132018494, −2.23770566603289254434521504446, −0.67248317442754312250359185687,
0.67248317442754312250359185687, 2.23770566603289254434521504446, 3.53143815977278326719132018494, 4.13717434369415984791877706804, 5.80434565975447127773988177363, 6.28306870048384612133156971537, 7.05773216835812314713426719636, 8.443038312056909725707071466074, 9.032837951737196726740129131865, 9.707726329741287678388099473700
