L(s) = 1 | − 1.61i·3-s + 0.618i·7-s + 0.381·9-s + 11-s − 2.23i·13-s + 4.85i·17-s − 5.47·19-s + 1.00·21-s + 6.32i·23-s − 5.47i·27-s + 4.38·29-s + 4.23·31-s − 1.61i·33-s + 1.76i·37-s − 3.61·39-s + ⋯ |
L(s) = 1 | − 0.934i·3-s + 0.233i·7-s + 0.127·9-s + 0.301·11-s − 0.620i·13-s + 1.17i·17-s − 1.25·19-s + 0.218·21-s + 1.31i·23-s − 1.05i·27-s + 0.813·29-s + 0.760·31-s − 0.281i·33-s + 0.289i·37-s − 0.579·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.986200411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.986200411\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 1.61iT - 3T^{2} \) |
| 7 | \( 1 - 0.618iT - 7T^{2} \) |
| 13 | \( 1 + 2.23iT - 13T^{2} \) |
| 17 | \( 1 - 4.85iT - 17T^{2} \) |
| 19 | \( 1 + 5.47T + 19T^{2} \) |
| 23 | \( 1 - 6.32iT - 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 - 1.76iT - 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 + 8.47iT - 43T^{2} \) |
| 47 | \( 1 - 4.70iT - 47T^{2} \) |
| 53 | \( 1 - 3.85iT - 53T^{2} \) |
| 59 | \( 1 + 3.76T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 0.291T + 71T^{2} \) |
| 73 | \( 1 - 7.09iT - 73T^{2} \) |
| 79 | \( 1 - 2.85T + 79T^{2} \) |
| 83 | \( 1 + 1.14iT - 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 4.90iT - 97T^{2} \) |
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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
−8.205796442452457157419460624079, −7.61565517178773584052599993314, −6.83699185398964113748031797998, −6.18765224022011256287300472988, −5.61793465377092512294985010723, −4.47058082058818953434870500407, −3.77181760930016226190049238723, −2.63081977474201185425563837749, −1.80272941827143324742554050250, −0.878963725863667209154053100124,
0.74266947111535914638791438992, 2.15501763515599214351969944865, 3.05240985370583536818617408536, 4.25948342233309574186124681337, 4.36983937248934180989892296927, 5.24236308919107146694887941317, 6.39403421370970930981547711590, 6.78925169381256009987895440908, 7.73085524899634670647339411479, 8.596744381961167545060230249091