L(s) = 1 | + 0.940i·5-s − 7-s − 5.98i·11-s − 6.59i·13-s + 2.64·17-s + 5.83i·19-s + 2.88·23-s + 4.11·25-s − 3.09i·29-s − 3.52·31-s − 0.940i·35-s − 0.213i·37-s + 1.63·41-s − 7.16i·43-s − 9.32·47-s + ⋯ |
L(s) = 1 | + 0.420i·5-s − 0.377·7-s − 1.80i·11-s − 1.82i·13-s + 0.640·17-s + 1.33i·19-s + 0.601·23-s + 0.823·25-s − 0.575i·29-s − 0.632·31-s − 0.158i·35-s − 0.0351i·37-s + 0.255·41-s − 1.09i·43-s − 1.36·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.214839079\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.214839079\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 0.940iT - 5T^{2} \) |
| 11 | \( 1 + 5.98iT - 11T^{2} \) |
| 13 | \( 1 + 6.59iT - 13T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 - 5.83iT - 19T^{2} \) |
| 23 | \( 1 - 2.88T + 23T^{2} \) |
| 29 | \( 1 + 3.09iT - 29T^{2} \) |
| 31 | \( 1 + 3.52T + 31T^{2} \) |
| 37 | \( 1 + 0.213iT - 37T^{2} \) |
| 41 | \( 1 - 1.63T + 41T^{2} \) |
| 43 | \( 1 + 7.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.32T + 47T^{2} \) |
| 53 | \( 1 - 7.51iT - 53T^{2} \) |
| 59 | \( 1 - 11.9iT - 59T^{2} \) |
| 61 | \( 1 - 1.48iT - 61T^{2} \) |
| 67 | \( 1 + 13.0iT - 67T^{2} \) |
| 71 | \( 1 + 1.54T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 8.74iT - 83T^{2} \) |
| 89 | \( 1 + 7.50T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.86428661715487366847813260970, −7.23370191222579187886287052707, −6.14516412590328490082566766370, −5.81509632248894515190770302770, −5.20350827626593313927213400767, −3.89823682150989750195452798968, −3.15480220115402865064214965875, −2.89118653480936011363180595510, −1.29671791650764008438797838401, −0.32676874169468905322602998586,
1.29120173278724428472512133604, 2.08886275384095098895541406835, 3.04356533318485907592833180354, 4.12889827641701406634066606293, 4.73512396667586381439972239547, 5.19528457235225505441215537254, 6.47719061386821230443415360359, 6.94052691910487303586633244763, 7.35807806449751873751689402205, 8.452826223651538342996908599749