L(s) = 1 | + 3-s + 3.20·5-s + 9-s − 4.24·11-s + 3.15·13-s + 3.20·15-s + 4.40·17-s + 3.15·19-s − 4.40·23-s + 5.24·25-s + 27-s + 7.20·29-s + 2.04·31-s − 4.24·33-s − 9.65·37-s + 3.15·39-s + 10.4·41-s − 0.750·43-s + 3.20·45-s − 2.40·47-s + 4.40·51-s − 3.29·53-s − 13.6·55-s + 3.15·57-s + 8.24·59-s + 8.09·61-s + 10.0·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.43·5-s + 0.333·9-s − 1.28·11-s + 0.874·13-s + 0.826·15-s + 1.06·17-s + 0.723·19-s − 0.918·23-s + 1.04·25-s + 0.192·27-s + 1.33·29-s + 0.367·31-s − 0.739·33-s − 1.58·37-s + 0.504·39-s + 1.63·41-s − 0.114·43-s + 0.477·45-s − 0.350·47-s + 0.616·51-s − 0.452·53-s − 1.83·55-s + 0.417·57-s + 1.07·59-s + 1.03·61-s + 1.25·65-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
\(L(1)\) |
\(\approx\) |
\(3.411595614\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.411595614\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.20T + 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 - 4.40T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 + 4.40T + 23T^{2} \) |
| 29 | \( 1 - 7.20T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 0.750T + 43T^{2} \) |
| 47 | \( 1 + 2.40T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 - 8.09T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 + 6.40T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 2.40T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−8.267414357177053185596952568660, −7.75105721640099127773769886330, −6.82593107590799135907221048444, −5.96012465133648246777793467390, −5.51646067411035830483502473946, −4.71863203383945951025678417052, −3.53535953043013448970352082283, −2.79601212502706247141468895745, −2.02081170243445436849726652818, −1.05453466129633628542132417669,
1.05453466129633628542132417669, 2.02081170243445436849726652818, 2.79601212502706247141468895745, 3.53535953043013448970352082283, 4.71863203383945951025678417052, 5.51646067411035830483502473946, 5.96012465133648246777793467390, 6.82593107590799135907221048444, 7.75105721640099127773769886330, 8.267414357177053185596952568660