L(s) = 1 | + (1 − i)2-s − 3-s − 2i·4-s + (2 − i)5-s + (−1 + i)6-s + i·7-s + (−2 − 2i)8-s + 9-s + (1 − 3i)10-s + 2i·12-s + 6·13-s + (1 + i)14-s + (−2 + i)15-s − 4·16-s − 2i·17-s + (1 − i)18-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 0.577·3-s − i·4-s + (0.894 − 0.447i)5-s + (−0.408 + 0.408i)6-s + 0.377i·7-s + (−0.707 − 0.707i)8-s + 0.333·9-s + (0.316 − 0.948i)10-s + 0.577i·12-s + 1.66·13-s + (0.267 + 0.267i)14-s + (−0.516 + 0.258i)15-s − 16-s − 0.485i·17-s + (0.235 − 0.235i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28300 - 1.78007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28300 - 1.78007i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.22055564903449068597804489576, −9.150034713544318670605475273109, −8.722047973217996408951500765928, −6.93524927378989757660266026624, −6.14640317875450093577799490037, −5.44113838224845248127704045827, −4.68967901835867317831478671084, −3.47741256638108773246855468454, −2.17890364115498778427599068916, −1.00520541145744263086135747723,
1.72965069967063217883851920422, 3.37809013706155145767483498617, 4.16222293778383568350588310650, 5.58888338268837600996760223866, 5.91943038126992156914162973044, 6.74687877622405378223541267975, 7.65494799383293181616186704631, 8.624863431129588132276124551128, 9.623661535681515484334411058219, 10.63855688308263044073699763356