L(s) = 1 | + (−1.14 + 1.30i)3-s + 5-s − 5.02i·7-s + (−0.396 − 2.97i)9-s + 4.49·11-s − 4.52i·13-s + (−1.14 + 1.30i)15-s + 1.07i·17-s + 2.27·19-s + (6.54 + 5.73i)21-s − 8.57i·23-s + 25-s + (4.32 + 2.87i)27-s − 1.56i·29-s + 8.54i·31-s + ⋯ |
L(s) = 1 | + (−0.658 + 0.752i)3-s + 0.447·5-s − 1.89i·7-s + (−0.132 − 0.991i)9-s + 1.35·11-s − 1.25i·13-s + (−0.294 + 0.336i)15-s + 0.261i·17-s + 0.522·19-s + (1.42 + 1.25i)21-s − 1.78i·23-s + 0.200·25-s + (0.832 + 0.553i)27-s − 0.290i·29-s + 1.53i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00711 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00711 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.609609300\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.609609300\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.14 - 1.30i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (6.11 - 5.43i)T \) |
good | 7 | \( 1 + 5.02iT - 7T^{2} \) |
| 11 | \( 1 - 4.49T + 11T^{2} \) |
| 13 | \( 1 + 4.52iT - 13T^{2} \) |
| 17 | \( 1 - 1.07iT - 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 + 8.57iT - 23T^{2} \) |
| 29 | \( 1 + 1.56iT - 29T^{2} \) |
| 31 | \( 1 - 8.54iT - 31T^{2} \) |
| 37 | \( 1 - 0.199T + 37T^{2} \) |
| 41 | \( 1 - 4.59T + 41T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 - 8.77iT - 47T^{2} \) |
| 53 | \( 1 - 2.42T + 53T^{2} \) |
| 59 | \( 1 + 12.5iT - 59T^{2} \) |
| 61 | \( 1 + 0.619iT - 61T^{2} \) |
| 71 | \( 1 - 5.87iT - 71T^{2} \) |
| 73 | \( 1 + 5.16T + 73T^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 + 4.51iT - 83T^{2} \) |
| 89 | \( 1 + 16.9iT - 89T^{2} \) |
| 97 | \( 1 + 13.3iT - 97T^{2} \) |
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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
−8.358342727588566183126464392996, −7.32259878372103539319489396614, −6.71455451835990650386573977833, −6.12552849520731838909970134276, −5.17463653406609639386007940314, −4.39187639798743030871035297961, −3.83388774106579140827731985401, −3.03163479940022297605352483490, −1.27791220048342099618166608638, −0.56367270236840311587686724289,
1.39059652288343333159886003357, 1.98642800034458298358826814276, 2.91264784003527524653625096422, 4.18793539729456649692409702901, 5.17811254028757836149761266941, 5.83452024807898123994859460361, 6.25053311211950209160964755372, 7.01327895439845175939709306856, 7.81589230890084635267663905463, 8.759761166876542460366198448456