| L(s) = 1 | − 1.73i·3-s + (−1.63 − 1.52i)5-s − 5.27·11-s − 2.62i·13-s + (−2.63 + 2.83i)15-s + 0.418i·17-s − 3.27·19-s + 7.82i·23-s + (0.362 + 4.98i)25-s − 5.19i·27-s − 4.27·29-s + 3.27·31-s + 9.13i·33-s − 9.97i·37-s − 4.54·39-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + (−0.732 − 0.680i)5-s − 1.59·11-s − 0.728i·13-s + (−0.680 + 0.732i)15-s + 0.101i·17-s − 0.751·19-s + 1.63i·23-s + (0.0725 + 0.997i)25-s − 1.00i·27-s − 0.793·29-s + 0.588·31-s + 1.59i·33-s − 1.63i·37-s − 0.728·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.122424 + 0.311423i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.122424 + 0.311423i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.63 + 1.52i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.73iT - 3T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 + 2.62iT - 13T^{2} \) |
| 17 | \( 1 - 0.418iT - 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 - 7.82iT - 23T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 + 9.97iT - 37T^{2} \) |
| 41 | \( 1 + 3.72T + 41T^{2} \) |
| 43 | \( 1 - 2.15iT - 43T^{2} \) |
| 47 | \( 1 - 6.50iT - 47T^{2} \) |
| 53 | \( 1 - 5.67iT - 53T^{2} \) |
| 59 | \( 1 - 3.27T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 3.52iT - 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 + 6.50iT - 73T^{2} \) |
| 79 | \( 1 + 7.27T + 79T^{2} \) |
| 83 | \( 1 - 7.40iT - 83T^{2} \) |
| 89 | \( 1 - 7T + 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−9.384066696427408896320986229769, −8.355961166120597213748209635776, −7.65354184175091055727912419306, −7.37940209413697101786663662767, −5.97895002338454314539707554362, −5.21747600458643885160863039927, −4.12936958292740178855440717374, −2.87246584958509110039018861677, −1.57719551285500936595430011296, −0.14982028937357912319505609570,
2.36985197177523433474671115118, 3.39068386288163115165096481636, 4.40872524139374407758221726370, 4.97758471321037013244516068047, 6.33452093959358487028270113513, 7.16299970214427679415899052316, 8.113536126278013440736777859913, 8.805116315245803231078958460782, 10.05749508215314717036121780905, 10.37546714804576652590577271595
