L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 4·11-s − 12-s + 13-s + 16-s + 18-s + 19-s + 4·22-s − 4·23-s − 24-s + 26-s − 27-s − 3·29-s + 4·31-s + 32-s − 4·33-s + 36-s + 5·37-s + 38-s − 39-s + 9·41-s − 2·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.235·18-s + 0.229·19-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.557·29-s + 0.718·31-s + 0.176·32-s − 0.696·33-s + 1/6·36-s + 0.821·37-s + 0.162·38-s − 0.160·39-s + 1.40·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.510833062\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.510833062\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.346294914617349227391665037891, −8.294562702533927607021993216132, −7.43567561529259377939578124644, −6.52271583617779153448154910349, −6.07141832466387942382967212478, −5.16082781440088906246161673826, −4.22411160569367787621331111626, −3.60917175599476632450565586646, −2.26942715625149527103295903448, −1.04682770281730633092487857306,
1.04682770281730633092487857306, 2.26942715625149527103295903448, 3.60917175599476632450565586646, 4.22411160569367787621331111626, 5.16082781440088906246161673826, 6.07141832466387942382967212478, 6.52271583617779153448154910349, 7.43567561529259377939578124644, 8.294562702533927607021993216132, 9.346294914617349227391665037891