| L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s + 4·13-s − 4·14-s + 16-s + 6·17-s − 2·19-s − 20-s − 6·23-s + 25-s − 4·26-s + 4·28-s + 6·29-s + 8·31-s − 32-s − 6·34-s − 4·35-s + 2·37-s + 2·38-s + 40-s + 6·41-s + 10·43-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s + 1.10·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.458·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s − 0.784·26-s + 0.755·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 1.02·34-s − 0.676·35-s + 0.328·37-s + 0.324·38-s + 0.158·40-s + 0.937·41-s + 1.52·43-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.073758644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.073758644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.42071807319542, −16.08057713416091, −15.46934103968118, −14.85094759881894, −14.22939480995220, −13.94603832233303, −13.02465876341383, −12.09710592495068, −11.91163527003461, −11.28083558870901, −10.60203124966908, −10.28037300346622, −9.432618012834167, −8.588301413191471, −8.222043707773533, −7.813768943649960, −7.204130191603557, −6.092337078757644, −5.852079924130861, −4.707531393111455, −4.242646315437053, −3.308184743000544, −2.399759980389422, −1.425395506675942, −0.8453637738747555,
0.8453637738747555, 1.425395506675942, 2.399759980389422, 3.308184743000544, 4.242646315437053, 4.707531393111455, 5.852079924130861, 6.092337078757644, 7.204130191603557, 7.813768943649960, 8.222043707773533, 8.588301413191471, 9.432618012834167, 10.28037300346622, 10.60203124966908, 11.28083558870901, 11.91163527003461, 12.09710592495068, 13.02465876341383, 13.94603832233303, 14.22939480995220, 14.85094759881894, 15.46934103968118, 16.08057713416091, 16.42071807319542
