L(s) = 1 | − 2.98·3-s + 2.28·5-s + 7-s + 5.89·9-s + 11-s − 13-s − 6.82·15-s + 7.65·17-s + 4.26·19-s − 2.98·21-s + 6.60·23-s + 0.237·25-s − 8.61·27-s + 4.02·29-s − 5.67·31-s − 2.98·33-s + 2.28·35-s + 2.12·37-s + 2.98·39-s + 5.12·41-s − 11.8·43-s + 13.4·45-s + 2.94·47-s + 49-s − 22.8·51-s + 14.4·53-s + 2.28·55-s + ⋯ |
L(s) = 1 | − 1.72·3-s + 1.02·5-s + 0.377·7-s + 1.96·9-s + 0.301·11-s − 0.277·13-s − 1.76·15-s + 1.85·17-s + 0.978·19-s − 0.650·21-s + 1.37·23-s + 0.0474·25-s − 1.65·27-s + 0.747·29-s − 1.01·31-s − 0.519·33-s + 0.386·35-s + 0.348·37-s + 0.477·39-s + 0.800·41-s − 1.80·43-s + 2.00·45-s + 0.429·47-s + 0.142·49-s − 3.19·51-s + 1.98·53-s + 0.308·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.603447705\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.603447705\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 - 4.26T + 19T^{2} \) |
| 23 | \( 1 - 6.60T + 23T^{2} \) |
| 29 | \( 1 - 4.02T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 37 | \( 1 - 2.12T + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 2.94T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 9.55T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 + 0.770T + 67T^{2} \) |
| 71 | \( 1 + 8.20T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 - 0.774T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
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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
−8.465457425481413327555067176062, −7.29897278859038572709500252745, −6.99435491641155693436537340136, −5.94410516328892843384404479017, −5.46200481548827574576104944051, −5.15686703383587233786958321385, −4.09439264216335295555014241726, −2.91402410924748651064377223770, −1.52639234232923929506575501988, −0.893294190739507390801075173165,
0.893294190739507390801075173165, 1.52639234232923929506575501988, 2.91402410924748651064377223770, 4.09439264216335295555014241726, 5.15686703383587233786958321385, 5.46200481548827574576104944051, 5.94410516328892843384404479017, 6.99435491641155693436537340136, 7.29897278859038572709500252745, 8.465457425481413327555067176062