L(s) = 1 | + i·3-s + 4i·7-s − 9-s − 4·11-s + 2i·13-s − 6i·17-s − 4·19-s − 4·21-s − i·27-s − 2·29-s − 4·31-s − 4i·33-s − 2i·37-s − 2·39-s + 2·41-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.51i·7-s − 0.333·9-s − 1.20·11-s + 0.554i·13-s − 1.45i·17-s − 0.917·19-s − 0.872·21-s − 0.192i·27-s − 0.371·29-s − 0.718·31-s − 0.696i·33-s − 0.328i·37-s − 0.320·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.758013295792964580189445896610, −8.175334856283479565184366458135, −7.18429845183093739347714495190, −6.25933822836416351247379396935, −5.28330711136303118673229284591, −5.05766962892158957405227478660, −3.82744557524133953554722235651, −2.69904648162715290918230431263, −2.16721683340337093038419781585, 0,
1.27153017758547989564993590404, 2.41997882473594672394905060754, 3.56822262297053512783743584258, 4.31489484676743874612318866459, 5.36560366892489293594598388830, 6.20534235550408897063443770371, 6.98253808718326664738695357366, 7.86098078024792670901306255406, 8.031608597980090646472922657602