| L(s) = 1 | − 3.04i·3-s + 4.30i·5-s + (−2.63 − 0.225i)7-s − 6.29·9-s + (2.23 + 2.45i)11-s + 13-s + 13.1·15-s + 6.22·17-s + 0.815·19-s + (−0.688 + 8.03i)21-s − 0.626·23-s − 13.5·25-s + 10.0i·27-s + 5.50i·29-s − 1.54i·31-s + ⋯ |
| L(s) = 1 | − 1.76i·3-s + 1.92i·5-s + (−0.996 − 0.0854i)7-s − 2.09·9-s + (0.672 + 0.740i)11-s + 0.277·13-s + 3.38·15-s + 1.51·17-s + 0.187·19-s + (−0.150 + 1.75i)21-s − 0.130·23-s − 2.70·25-s + 1.93i·27-s + 1.02i·29-s − 0.277i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4427245832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4427245832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.63 + 0.225i)T \) |
| 11 | \( 1 + (-2.23 - 2.45i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.04iT - 3T^{2} \) |
| 5 | \( 1 - 4.30iT - 5T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 - 0.815T + 19T^{2} \) |
| 23 | \( 1 + 0.626T + 23T^{2} \) |
| 29 | \( 1 - 5.50iT - 29T^{2} \) |
| 31 | \( 1 + 1.54iT - 31T^{2} \) |
| 37 | \( 1 + 5.94T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 2.96iT - 43T^{2} \) |
| 47 | \( 1 + 10.6iT - 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 7.99iT - 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 - 6.46T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 9.52iT - 79T^{2} \) |
| 83 | \( 1 + 8.25T + 83T^{2} \) |
| 89 | \( 1 + 1.67iT - 89T^{2} \) |
| 97 | \( 1 + 3.51iT - 97T^{2} \) |
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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
−8.399809231867270567759537143062, −7.61430674400942986661747736930, −7.11139418800894863973368768426, −6.62795982204640565813710085947, −6.25731305153411625906352382488, −5.36668924061618172495450172564, −3.45334511763292067965532549936, −3.32924910845673315571189882700, −2.26495956615905943486372872693, −1.44082369967412560413925687663,
0.13039293619658600995778494750, 1.33334023845741674463793477625, 3.12678044381389034969182061672, 3.68569738103448155947365860335, 4.35723404875063947361205842158, 5.16257801292639354012160317399, 5.66244551940476678700620111287, 6.32130293349840343363813508568, 7.87910792829927920579550847690, 8.465385671512852126594964353596
