| L(s) = 1 | − 0.671i·3-s − 1.59i·5-s + (1.51 + 2.17i)7-s + 2.54·9-s + (−1.91 + 2.71i)11-s − 13-s − 1.07·15-s − 5.43·17-s − 1.69·19-s + (1.45 − 1.01i)21-s + 6.70·23-s + 2.45·25-s − 3.72i·27-s − 1.43i·29-s − 4.27i·31-s + ⋯ |
| L(s) = 1 | − 0.387i·3-s − 0.713i·5-s + (0.571 + 0.820i)7-s + 0.849·9-s + (−0.575 + 0.817i)11-s − 0.277·13-s − 0.276·15-s − 1.31·17-s − 0.389·19-s + (0.317 − 0.221i)21-s + 1.39·23-s + 0.491·25-s − 0.716i·27-s − 0.266i·29-s − 0.768i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.032301282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.032301282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.51 - 2.17i)T \) |
| 11 | \( 1 + (1.91 - 2.71i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.671iT - 3T^{2} \) |
| 5 | \( 1 + 1.59iT - 5T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 + 1.43iT - 29T^{2} \) |
| 31 | \( 1 + 4.27iT - 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 - 1.78iT - 43T^{2} \) |
| 47 | \( 1 - 6.96iT - 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 2.50iT - 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 - 9.10T + 71T^{2} \) |
| 73 | \( 1 + 3.96T + 73T^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 7.61iT - 89T^{2} \) |
| 97 | \( 1 + 13.4iT - 97T^{2} \) |
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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
−8.441013362581899267036094933153, −7.74396404795335539474988933627, −7.02603802965488233553680164229, −6.34879011475524050862096203935, −5.24741849060737995042049827976, −4.76734057963672627683673159301, −4.13881849744844456984275455022, −2.57597282382676060235956536803, −2.04079758759246176321631350244, −0.926122211801789955796740129290,
0.74661519197953626103060222515, 2.03570096651724453224404803324, 3.05528620764636883531329392834, 3.86276140442886405217358433493, 4.69870178036602835134868725322, 5.23359009267397251533281289747, 6.48959213590423928904166682014, 7.01119503459644009466671166353, 7.53370086198037035057630454576, 8.579431034627280865769523245099
