L(s) = 1 | − 2·7-s − 3·9-s − 4·11-s − 2·13-s + 2·17-s + 19-s + 2·23-s − 5·25-s − 2·29-s − 8·31-s + 10·37-s + 6·41-s + 4·43-s − 2·47-s − 3·49-s − 2·53-s − 12·59-s + 8·61-s + 6·63-s − 4·67-s + 12·71-s + 10·73-s + 8·77-s + 4·79-s + 9·81-s − 4·83-s − 6·89-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s + 0.417·23-s − 25-s − 0.371·29-s − 1.43·31-s + 1.64·37-s + 0.937·41-s + 0.609·43-s − 0.291·47-s − 3/7·49-s − 0.274·53-s − 1.56·59-s + 1.02·61-s + 0.755·63-s − 0.488·67-s + 1.42·71-s + 1.17·73-s + 0.911·77-s + 0.450·79-s + 81-s − 0.439·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8977293064\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8977293064\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.973068880103887975533099453194, −7.78696955835700262631490859442, −6.85650456352980196839664850224, −5.86139423474072002063661239793, −5.56660148839687596241765386210, −4.64774004103704763608405557135, −3.55640808726731998436678026075, −2.88455777627499735444003252682, −2.13653116379609338150525447154, −0.48714338876172827636734993514,
0.48714338876172827636734993514, 2.13653116379609338150525447154, 2.88455777627499735444003252682, 3.55640808726731998436678026075, 4.64774004103704763608405557135, 5.56660148839687596241765386210, 5.86139423474072002063661239793, 6.85650456352980196839664850224, 7.78696955835700262631490859442, 7.973068880103887975533099453194