L(s) = 1 | − 1.29i·5-s + i·7-s − 2.21·11-s − 1.03·13-s − 2.24i·17-s − 8.63i·19-s + 8.96·23-s + 3.31·25-s + 3.26i·29-s − 4.37i·31-s + 1.29·35-s − 7.21·37-s − 7.58i·41-s + 12.9i·43-s − 5.24·47-s + ⋯ |
L(s) = 1 | − 0.579i·5-s + 0.377i·7-s − 0.668·11-s − 0.287·13-s − 0.544i·17-s − 1.98i·19-s + 1.87·23-s + 0.663·25-s + 0.606i·29-s − 0.786i·31-s + 0.219·35-s − 1.18·37-s − 1.18i·41-s + 1.97i·43-s − 0.764·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.075323350\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075323350\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 5 | \( 1 + 1.29iT - 5T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 + 1.03T + 13T^{2} \) |
| 17 | \( 1 + 2.24iT - 17T^{2} \) |
| 19 | \( 1 + 8.63iT - 19T^{2} \) |
| 23 | \( 1 - 8.96T + 23T^{2} \) |
| 29 | \( 1 - 3.26iT - 29T^{2} \) |
| 31 | \( 1 + 4.37iT - 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 + 7.58iT - 41T^{2} \) |
| 43 | \( 1 - 12.9iT - 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 - 11.1iT - 53T^{2} \) |
| 59 | \( 1 - 2.46T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 4.46iT - 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 9.14iT - 79T^{2} \) |
| 83 | \( 1 - 9.96T + 83T^{2} \) |
| 89 | \( 1 - 3.48iT - 89T^{2} \) |
| 97 | \( 1 + 6.30T + 97T^{2} \) |
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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
−7.74657865550751761078556638975, −7.11297181874972611219095261457, −6.52649010485192274071932572596, −5.36065702743579594952016082901, −5.01842171791802164630730296433, −4.41579510124662145172317335687, −2.98013145392206223436608203683, −2.71328894739271697873788421815, −1.35494403204187320746342895214, −0.28641693946923995909113954676,
1.22857444067921497098430266054, 2.23012232839742621915700347366, 3.25945959454023476091665569336, 3.71005801171819039938516385177, 4.88930608032903979115990502881, 5.37033772361515534607278577891, 6.37889710539491762717867929399, 6.89109913120403543389292965605, 7.63317125299786997944132903284, 8.243478282698189222126197181789