L(s) = 1 | − 5-s − 7-s − 3·9-s + 2·11-s + 13-s + 8·19-s − 8·23-s + 25-s + 6·29-s − 10·31-s + 35-s + 2·37-s + 10·43-s + 3·45-s − 8·47-s + 49-s − 12·53-s − 2·55-s + 6·61-s + 3·63-s − 65-s + 4·71-s − 6·73-s − 2·77-s − 16·79-s + 9·81-s − 4·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s + 0.603·11-s + 0.277·13-s + 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 1.79·31-s + 0.169·35-s + 0.328·37-s + 1.52·43-s + 0.447·45-s − 1.16·47-s + 1/7·49-s − 1.64·53-s − 0.269·55-s + 0.768·61-s + 0.377·63-s − 0.124·65-s + 0.474·71-s − 0.702·73-s − 0.227·77-s − 1.80·79-s + 81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.101917935357030907670183424417, −7.56025189958285642546063172694, −6.64828491987661026865108105537, −5.90351247393730279741713014317, −5.28651403654828433914025679312, −4.16405444916472321316719134794, −3.45627384662986438687428913796, −2.69377150022178809869710718931, −1.35472812286248210433196729500, 0,
1.35472812286248210433196729500, 2.69377150022178809869710718931, 3.45627384662986438687428913796, 4.16405444916472321316719134794, 5.28651403654828433914025679312, 5.90351247393730279741713014317, 6.64828491987661026865108105537, 7.56025189958285642546063172694, 8.101917935357030907670183424417