L(s) = 1 | − 2.75·3-s − 1.98i·5-s − i·7-s + 4.60·9-s + i·11-s + (−2.54 − 2.55i)13-s + 5.46i·15-s − 7.39·17-s + 4.19i·19-s + 2.75i·21-s + 4.61·23-s + 1.07·25-s − 4.42·27-s + 4.07·29-s − 3.60i·31-s + ⋯ |
L(s) = 1 | − 1.59·3-s − 0.886i·5-s − 0.377i·7-s + 1.53·9-s + 0.301i·11-s + (−0.704 − 0.709i)13-s + 1.41i·15-s − 1.79·17-s + 0.963i·19-s + 0.601i·21-s + 0.963·23-s + 0.214·25-s − 0.851·27-s + 0.755·29-s − 0.647i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7589341226\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7589341226\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (2.54 + 2.55i)T \) |
good | 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 + 1.98iT - 5T^{2} \) |
| 17 | \( 1 + 7.39T + 17T^{2} \) |
| 19 | \( 1 - 4.19iT - 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 + 3.60iT - 31T^{2} \) |
| 37 | \( 1 - 9.67iT - 37T^{2} \) |
| 41 | \( 1 - 5.43iT - 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 - 0.284iT - 47T^{2} \) |
| 53 | \( 1 + 4.46T + 53T^{2} \) |
| 59 | \( 1 - 1.96iT - 59T^{2} \) |
| 61 | \( 1 + 2.69T + 61T^{2} \) |
| 67 | \( 1 + 12.8iT - 67T^{2} \) |
| 71 | \( 1 - 8.00iT - 71T^{2} \) |
| 73 | \( 1 - 0.542iT - 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 3.32iT - 83T^{2} \) |
| 89 | \( 1 - 14.4iT - 89T^{2} \) |
| 97 | \( 1 - 15.3iT - 97T^{2} \) |
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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
−8.307380595000448237163741983019, −7.50282257349179754923266273936, −6.68615625428160022587882227130, −6.15138317766376104052429261194, −5.25453907472986679043777793838, −4.70054215378723134728691104004, −4.25815694804899556504419280001, −2.76869990674786671753525445031, −1.41262363932963213911395364180, −0.52380652954286586927221298749,
0.58580232054965776565407735544, 2.12375416461109084408972161183, 2.93372325221350235851785838875, 4.33662154117921168679096796162, 4.78647300272984355470456241894, 5.68272654753363561829129196255, 6.36056056117958519326354831317, 7.02305187102446154595774810929, 7.24986541826764479471937054046, 8.846739205427400617503898138938