L(s) = 1 | − i·2-s + 2i·3-s − 4-s + 2·6-s + i·8-s − 9-s − 4·11-s − 2i·12-s + 2i·13-s + 16-s − 8i·17-s + i·18-s + 6·19-s + 4i·22-s − 4i·23-s − 2·24-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.15i·3-s − 0.5·4-s + 0.816·6-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s − 0.577i·12-s + 0.554i·13-s + 0.250·16-s − 1.94i·17-s + 0.235i·18-s + 1.37·19-s + 0.852i·22-s − 0.834i·23-s − 0.408·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.587991641\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.587991641\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.309715555082081133316007707851, −8.476377517317268353250078622555, −7.60256676112824481143366506774, −6.72428270293329093466002835358, −5.39208242529338612975099560943, −4.86952389118626788525970680761, −4.30483223082779562216024553941, −3.05145050008736289448953964345, −2.66175036468782064581884762787, −0.924637238044766700621084998264,
0.72364501131582955918842644560, 1.89360482654276718316681218117, 3.04981496558464750891819275428, 4.13077031631429424517547442500, 5.34131717998721515322648589431, 5.80978254493715340456863322811, 6.68470421017370249373208073045, 7.42266567160512747634215248184, 8.014243131926179487228188988593, 8.381995041853395292428856848994