L(s) = 1 | + (0.390 − 1.35i)2-s + (−1.69 − 1.06i)4-s − i·5-s + 1.67i·7-s + (−2.10 + 1.89i)8-s + (−1.35 − 0.390i)10-s + 4.67·11-s − 0.142·13-s + (2.27 + 0.653i)14-s + (1.74 + 3.59i)16-s + 7.09i·17-s + 0.158i·19-s + (−1.06 + 1.69i)20-s + (1.82 − 6.35i)22-s − 0.185·23-s + ⋯ |
L(s) = 1 | + (0.276 − 0.961i)2-s + (−0.847 − 0.530i)4-s − 0.447i·5-s + 0.632i·7-s + (−0.743 + 0.668i)8-s + (−0.429 − 0.123i)10-s + 1.41·11-s − 0.0394·13-s + (0.607 + 0.174i)14-s + (0.436 + 0.899i)16-s + 1.72i·17-s + 0.0363i·19-s + (−0.237 + 0.379i)20-s + (0.389 − 1.35i)22-s − 0.0386·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.783970433\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783970433\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.390 + 1.35i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 - 1.67iT - 7T^{2} \) |
| 11 | \( 1 - 4.67T + 11T^{2} \) |
| 13 | \( 1 + 0.142T + 13T^{2} \) |
| 17 | \( 1 - 7.09iT - 17T^{2} \) |
| 19 | \( 1 - 0.158iT - 19T^{2} \) |
| 23 | \( 1 + 0.185T + 23T^{2} \) |
| 29 | \( 1 - 3.22iT - 29T^{2} \) |
| 31 | \( 1 - 5.91iT - 31T^{2} \) |
| 37 | \( 1 - 0.634T + 37T^{2} \) |
| 41 | \( 1 - 10.8iT - 41T^{2} \) |
| 43 | \( 1 + 8.39iT - 43T^{2} \) |
| 47 | \( 1 - 9.55T + 47T^{2} \) |
| 53 | \( 1 + 7.27iT - 53T^{2} \) |
| 59 | \( 1 + 6.46T + 59T^{2} \) |
| 61 | \( 1 - 6.27T + 61T^{2} \) |
| 67 | \( 1 + 8.44iT - 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 4.30T + 73T^{2} \) |
| 79 | \( 1 + 13.8iT - 79T^{2} \) |
| 83 | \( 1 + 6.56T + 83T^{2} \) |
| 89 | \( 1 - 11.0iT - 89T^{2} \) |
| 97 | \( 1 - 3.53T + 97T^{2} \) |
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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
−9.250324765718903363166099446149, −8.813874619498667990853624162284, −8.110071599950466829815361203678, −6.64934907967577559113020033569, −5.92998579923209212404545298041, −5.05703740044127135320671390098, −4.08974474673026368319075407761, −3.42332463590397794144343211278, −2.07043083772771094184483037495, −1.20868738052474997152170351783,
0.75100456430238863518551393820, 2.67018597895236603073324875099, 3.85290813387658004910814901205, 4.38388929247216665994527761768, 5.52731289379553171089254425054, 6.33666010056644459269258818036, 7.16618771256565522479412592733, 7.47650381427568485975567665113, 8.632162260750469050052960060457, 9.393387023193395555646645473157