L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 4·11-s + 2·13-s + 15-s + 6·17-s + 4·19-s + 21-s − 23-s + 25-s − 27-s + 6·29-s − 8·31-s − 4·33-s + 35-s + 6·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s + 49-s − 6·51-s − 10·53-s − 4·55-s − 4·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 1.37·53-s − 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.359153245\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.359153245\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.64950678484152, −14.32923022645906, −13.97920036200965, −13.09618169227274, −12.62124658914550, −12.23428403155314, −11.58190535742317, −11.35210422280178, −10.71333775810350, −9.988643503867096, −9.574564398645474, −9.132377574348403, −8.299272905575972, −7.832570185681552, −7.217778534091722, −6.607194417439247, −6.149557267540288, −5.518535616811076, −4.956371249544373, −4.149079766412096, −3.569257932182414, −3.208391938822395, −2.078895582612064, −1.120378364569203, −0.7185259607642632,
0.7185259607642632, 1.120378364569203, 2.078895582612064, 3.208391938822395, 3.569257932182414, 4.149079766412096, 4.956371249544373, 5.518535616811076, 6.149557267540288, 6.607194417439247, 7.217778534091722, 7.832570185681552, 8.299272905575972, 9.132377574348403, 9.574564398645474, 9.988643503867096, 10.71333775810350, 11.35210422280178, 11.58190535742317, 12.23428403155314, 12.62124658914550, 13.09618169227274, 13.97920036200965, 14.32923022645906, 14.64950678484152