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
\(L(1)\) |
\(\approx\) |
\(0.546264 - 1.38958i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.546264 - 1.38958i\) |
\(L(\frac{3}{2})\) |
|
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
−9.844755192134472921899048714148, −8.619801764797419695832760086512, −8.001113630154357454631438207766, −7.25394571343315023491584726957, −6.21704850716647199715687633832, −5.43388114629236229923074894758, −4.61451230414475539292835242962, −2.91229851661542080492586322657, −1.97147768849012565373514973959, −0.67009093197832453107159192149,
1.94149152314632860883411906680, 3.09914533081311057246613997630, 4.03944148425413741050640058134, 5.31589916978674769057325598316, 5.64157732044416580378530754101, 7.12266627363024209208745554155, 7.60352596594479196918412155374, 9.039413915953197922333230846196, 9.606647125399777283402477293828, 10.23526780595824844845959393662