L(s) = 1 | − 4·11-s + 6i·13-s − 6i·17-s − 4·19-s − 2·29-s + 8·31-s + 2i·37-s + 6·41-s − 12i·43-s − 8i·47-s + 7·49-s − 6i·53-s − 12·59-s + 14·61-s + 4i·67-s + ⋯ |
L(s) = 1 | − 1.20·11-s + 1.66i·13-s − 1.45i·17-s − 0.917·19-s − 0.371·29-s + 1.43·31-s + 0.328i·37-s + 0.937·41-s − 1.82i·43-s − 1.16i·47-s + 49-s − 0.824i·53-s − 1.56·59-s + 1.79·61-s + 0.488i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.246011035\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.246011035\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 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
−8.546914401518107204058978036954, −7.57694521132852757032869702172, −6.99634290600883530296573979046, −6.28176010398922427641314122390, −5.27850768104958722113804489999, −4.66299721486102806706325159265, −3.83782041939156324691646444102, −2.64794761222461551270902274564, −2.02409094623636756180326573389, −0.43233629985980295369211591990,
0.960396513874823235630431002852, 2.35982320578521408068811531528, 3.03701392698589328992365024720, 4.08953771212554761910014870521, 4.90529113699697236846665615401, 5.83626622806603151858225137821, 6.19701608648906102378272823166, 7.40295750780192372897750411817, 8.141260748405405423643749122721, 8.301108504865396915362581905499