L(s) = 1 | + (−0.0820 − 0.0473i)3-s + (0.413 + 0.716i)7-s + (−1.49 − 2.59i)9-s + (1.5 + 0.866i)11-s + (−1.40 − 3.32i)13-s + (−1.24 + 0.716i)17-s + (0.926 − 0.534i)19-s − 0.0783i·21-s + (−2.67 − 1.54i)23-s + 0.567i·27-s + (3.72 − 6.45i)29-s − 5.84i·31-s + (−0.0820 − 0.142i)33-s + (−0.491 + 0.851i)37-s + (−0.0424 + 0.339i)39-s + ⋯ |
L(s) = 1 | + (−0.0473 − 0.0273i)3-s + (0.156 + 0.270i)7-s + (−0.498 − 0.863i)9-s + (0.452 + 0.261i)11-s + (−0.388 − 0.921i)13-s + (−0.300 + 0.173i)17-s + (0.212 − 0.122i)19-s − 0.0171i·21-s + (−0.557 − 0.321i)23-s + 0.109i·27-s + (0.692 − 1.19i)29-s − 1.04i·31-s + (−0.0142 − 0.0247i)33-s + (−0.0808 + 0.140i)37-s + (−0.00680 + 0.0543i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0518 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0518 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.199975288\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.199975288\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (1.40 + 3.32i)T \) |
good | 3 | \( 1 + (0.0820 + 0.0473i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.413 - 0.716i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.24 - 0.716i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.926 + 0.534i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.67 + 1.54i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.72 + 6.45i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.84iT - 31T^{2} \) |
| 37 | \( 1 + (0.491 - 0.851i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.69 + 2.13i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (8.26 - 4.77i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 0.334iT - 53T^{2} \) |
| 59 | \( 1 + (-9.98 + 5.76i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.35 + 2.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.87 + 11.9i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.46 + 4.88i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 0.252T + 79T^{2} \) |
| 83 | \( 1 + 5.67T + 83T^{2} \) |
| 89 | \( 1 + (3.98 + 2.29i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.76 - 8.25i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.566271986091752948413494492333, −8.537290670529568344696286600930, −7.991321867086713439525341187555, −6.85805423640665175626805128431, −6.14648906865127404102269222251, −5.30748279498656951529297945236, −4.24075595123390294780679591691, −3.24321044062906512530647808284, −2.15088238698401561511076746246, −0.50638658977768569345835081980,
1.48623815818085756777760600428, 2.66599509450046246079130772969, 3.85320743139254781034987795296, 4.82046786858116777570410152792, 5.57392121074333715764764597482, 6.72257781495767999862709820824, 7.30013590552317735577422411352, 8.415285450516491757017551129317, 8.877001061375964416309128448870, 9.960917081272048779025924318546