| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3.37·7-s − 8-s + 9-s − 1.37·11-s − 12-s − 2·13-s + 3.37·14-s + 16-s + 17-s − 18-s + 3.37·19-s + 3.37·21-s + 1.37·22-s − 4.37·23-s + 24-s + 2·26-s − 27-s − 3.37·28-s − 2.74·29-s − 8.11·31-s − 32-s + 1.37·33-s − 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.27·7-s − 0.353·8-s + 0.333·9-s − 0.413·11-s − 0.288·12-s − 0.554·13-s + 0.901·14-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.773·19-s + 0.735·21-s + 0.292·22-s − 0.911·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.637·28-s − 0.509·29-s − 1.45·31-s − 0.176·32-s + 0.238·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5354420449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5354420449\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 23 | \( 1 + 4.37T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 8.11T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 - 7.37T + 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 + 7.11T + 59T^{2} \) |
| 61 | \( 1 - 9.11T + 61T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 - 4.37T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 - 8.74T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
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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
−9.160083140834387719937769567555, −8.047261086459208850021784057155, −7.33669877379976829630364242939, −6.72788400209651612664774457365, −5.85156969762441601096339750851, −5.27667623664469658032907857239, −3.95164368561694314257175396845, −3.08624244127148540310764745415, −1.99609609975692426308072017560, −0.50573883662229611494273106057,
0.50573883662229611494273106057, 1.99609609975692426308072017560, 3.08624244127148540310764745415, 3.95164368561694314257175396845, 5.27667623664469658032907857239, 5.85156969762441601096339750851, 6.72788400209651612664774457365, 7.33669877379976829630364242939, 8.047261086459208850021784057155, 9.160083140834387719937769567555
