L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 3·11-s + 3·13-s − 15-s + 17-s − 6·19-s − 21-s − 2·23-s − 4·25-s + 27-s − 6·29-s + 4·31-s − 3·33-s + 35-s + 11·37-s + 3·39-s + 12·41-s − 3·43-s − 45-s + 12·47-s + 49-s + 51-s − 5·53-s + 3·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.832·13-s − 0.258·15-s + 0.242·17-s − 1.37·19-s − 0.218·21-s − 0.417·23-s − 4/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.522·33-s + 0.169·35-s + 1.80·37-s + 0.480·39-s + 1.87·41-s − 0.457·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.140·51-s − 0.686·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−15.66087887309730, −15.38094884825137, −14.65593336121239, −14.21920817750323, −13.45341041862640, −13.12643436229254, −12.63963007916353, −12.07562253502179, −11.20923571576731, −10.91104579815153, −10.28149227049009, −9.579202823721551, −9.191636471914927, −8.337783373195654, −8.018090837407073, −7.532645902520769, −6.771436724937658, −5.967677953790337, −5.712867373120653, −4.531103966769734, −4.106263330003355, −3.523195836459961, −2.619116118583943, −2.185200029369879, −1.035229977024495, 0,
1.035229977024495, 2.185200029369879, 2.619116118583943, 3.523195836459961, 4.106263330003355, 4.531103966769734, 5.712867373120653, 5.967677953790337, 6.771436724937658, 7.532645902520769, 8.018090837407073, 8.337783373195654, 9.191636471914927, 9.579202823721551, 10.28149227049009, 10.91104579815153, 11.20923571576731, 12.07562253502179, 12.63963007916353, 13.12643436229254, 13.45341041862640, 14.21920817750323, 14.65593336121239, 15.38094884825137, 15.66087887309730