L(s) = 1 | + 3-s − 2·5-s − 7-s − 2·9-s − 2·11-s − 13-s − 2·15-s − 6·19-s − 21-s + 23-s − 25-s − 5·27-s + 29-s + 31-s − 2·33-s + 2·35-s − 6·37-s − 39-s + 3·41-s + 4·45-s − 3·47-s + 49-s + 6·53-s + 4·55-s − 6·57-s + 8·59-s − 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s − 0.516·15-s − 1.37·19-s − 0.218·21-s + 0.208·23-s − 1/5·25-s − 0.962·27-s + 0.185·29-s + 0.179·31-s − 0.348·33-s + 0.338·35-s − 0.986·37-s − 0.160·39-s + 0.468·41-s + 0.596·45-s − 0.437·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s − 0.794·57-s + 1.04·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−10.16673786677166160632666004109, −9.070779974501057520193215638534, −8.358613155308557717295477598428, −7.69056077249759299311358239103, −6.67304262446929603948637296660, −5.55189078154822193747255296332, −4.33787474248905925419942938484, −3.35801784679235957433166551955, −2.33715438185105929984655086773, 0,
2.33715438185105929984655086773, 3.35801784679235957433166551955, 4.33787474248905925419942938484, 5.55189078154822193747255296332, 6.67304262446929603948637296660, 7.69056077249759299311358239103, 8.358613155308557717295477598428, 9.070779974501057520193215638534, 10.16673786677166160632666004109