L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 4·7-s − 8-s + 9-s − 11-s − 2·12-s + 5·13-s + 4·14-s + 16-s − 18-s − 7·19-s + 8·21-s + 22-s + 3·23-s + 2·24-s − 5·26-s + 4·27-s − 4·28-s + 3·29-s + 5·31-s − 32-s + 2·33-s + 36-s − 4·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.577·12-s + 1.38·13-s + 1.06·14-s + 1/4·16-s − 0.235·18-s − 1.60·19-s + 1.74·21-s + 0.213·22-s + 0.625·23-s + 0.408·24-s − 0.980·26-s + 0.769·27-s − 0.755·28-s + 0.557·29-s + 0.898·31-s − 0.176·32-s + 0.348·33-s + 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4822323460\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4822323460\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−10.74521564615271351083532625872, −10.11570697793034251967591157049, −9.060629736692727982052143009658, −8.308432311993429052776895796471, −6.83867050556930259584502489599, −6.34789084561451547279207756635, −5.62751372928592451091194364685, −4.07426843716282602437429344140, −2.73583703894348633191519993386, −0.70583551721755832098515922904,
0.70583551721755832098515922904, 2.73583703894348633191519993386, 4.07426843716282602437429344140, 5.62751372928592451091194364685, 6.34789084561451547279207756635, 6.83867050556930259584502489599, 8.308432311993429052776895796471, 9.060629736692727982052143009658, 10.11570697793034251967591157049, 10.74521564615271351083532625872