L(s) = 1 | + 2i·3-s + 2i·5-s − 4·7-s − 9-s + 2i·11-s − 2i·13-s − 4·15-s − 2·17-s + 2i·19-s − 8i·21-s + 4·23-s + 25-s + 4i·27-s + 6i·29-s − 4·33-s + ⋯ |
L(s) = 1 | + 1.15i·3-s + 0.894i·5-s − 1.51·7-s − 0.333·9-s + 0.603i·11-s − 0.554i·13-s − 1.03·15-s − 0.485·17-s + 0.458i·19-s − 1.74i·21-s + 0.834·23-s + 0.200·25-s + 0.769i·27-s + 1.11i·29-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.371714 + 0.897397i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.371714 + 0.897397i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 14iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−12.46448586956374693128491949424, −11.05675814829446524247374047496, −10.31055245362243336087047752436, −9.776138170650733784260645661580, −8.851882489860000138251155333534, −7.20786807714381711280566941520, −6.44976683667966550556096328535, −5.08762159732160482445369213130, −3.73740289547189202194632871913, −2.90739036926013457245631156495,
0.77554292563920746309047607113, 2.60239244415137240943522716091, 4.21895121524273615476920088960, 5.83522495830210206987552686355, 6.66103041189581929330436903728, 7.56515021467950703892918779460, 8.870868915425085523508767953007, 9.423884709811879950950983404496, 10.82899110790509360137108028749, 12.01993603461779646561261081518