L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 4·11-s + 6·13-s + 16-s − 2·17-s + 4·19-s − 20-s − 4·22-s + 8·23-s + 25-s − 6·26-s − 6·29-s − 31-s − 32-s + 2·34-s − 2·37-s − 4·38-s + 40-s − 10·41-s − 4·43-s + 4·44-s − 8·46-s − 7·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s − 1.17·26-s − 1.11·29-s − 0.179·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.436678693\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.436678693\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.835422004383418714792203517873, −8.275089086557225812492554714091, −7.20337711477419209006519780082, −6.77221208182826147740860397687, −5.92023882042501964215979667322, −4.94765971227712432981373790656, −3.73755209254899201720632004479, −3.29722400490686987326042278475, −1.75521926372478427674210187159, −0.885900235613335794880294909795,
0.885900235613335794880294909795, 1.75521926372478427674210187159, 3.29722400490686987326042278475, 3.73755209254899201720632004479, 4.94765971227712432981373790656, 5.92023882042501964215979667322, 6.77221208182826147740860397687, 7.20337711477419209006519780082, 8.275089086557225812492554714091, 8.835422004383418714792203517873