L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 9-s − 11-s + 4·13-s + 4·15-s + 4·17-s − 2·21-s + 4·23-s − 25-s + 4·27-s − 6·29-s − 10·31-s + 2·33-s − 2·35-s − 6·37-s − 8·39-s + 4·41-s − 12·43-s − 2·45-s + 10·47-s + 49-s − 8·51-s − 6·53-s + 2·55-s − 2·59-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.03·15-s + 0.970·17-s − 0.436·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s − 1.79·31-s + 0.348·33-s − 0.338·35-s − 0.986·37-s − 1.28·39-s + 0.624·41-s − 1.82·43-s − 0.298·45-s + 1.45·47-s + 1/7·49-s − 1.12·51-s − 0.824·53-s + 0.269·55-s − 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−9.275951467887346935408038775382, −8.401164460248213891696224277574, −7.60485396777068502870784654080, −6.83793327932833057571081045155, −5.70015900664701843854698011684, −5.30138461732919402727155679326, −4.11323776412142174518703250626, −3.26984949414789173092793201853, −1.43444095024382319596213609185, 0,
1.43444095024382319596213609185, 3.26984949414789173092793201853, 4.11323776412142174518703250626, 5.30138461732919402727155679326, 5.70015900664701843854698011684, 6.83793327932833057571081045155, 7.60485396777068502870784654080, 8.401164460248213891696224277574, 9.275951467887346935408038775382