L(s) = 1 | − i·2-s + 4-s − 3i·8-s + 11-s − 2i·13-s − 16-s − 6i·17-s + 4·19-s − i·22-s + 4i·23-s − 2·26-s + 6·29-s − 8·31-s − 5i·32-s − 6·34-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.5·4-s − 1.06i·8-s + 0.301·11-s − 0.554i·13-s − 0.250·16-s − 1.45i·17-s + 0.917·19-s − 0.213i·22-s + 0.834i·23-s − 0.392·26-s + 1.11·29-s − 1.43·31-s − 0.883i·32-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.124811318\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.124811318\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.009981792517041325618127236724, −7.59448482337773441748506132355, −7.39069179412371329518447272914, −6.37493324228720213680640761161, −5.53687980139488326604143932450, −4.63119664790311578158452463125, −3.44263449373654366533492299072, −2.94094632208522213338571995682, −1.81554396880746257608592286178, −0.71585547468055594752250145745,
1.41628377244930702650415001326, 2.42401908289579383628510698703, 3.54432749584742144635558726392, 4.51752406140848018897844123689, 5.52503670452864334856020834340, 6.14759764385075023815049053321, 6.91492921967781475784808351504, 7.45389843172807523346412129344, 8.483163162409989331485370807124, 8.774782573549075341468635274820