L(s) = 1 | + 5-s + 3·7-s − 5·11-s − 13-s + 3·17-s − 8·19-s + 3·23-s + 25-s + 2·29-s − 10·31-s + 3·35-s − 3·37-s − 3·41-s − 10·43-s + 6·47-s + 2·49-s − 5·53-s − 5·55-s − 4·59-s − 7·61-s − 65-s + 12·67-s − 5·71-s + 2·73-s − 15·77-s − 13·79-s + 12·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.13·7-s − 1.50·11-s − 0.277·13-s + 0.727·17-s − 1.83·19-s + 0.625·23-s + 1/5·25-s + 0.371·29-s − 1.79·31-s + 0.507·35-s − 0.493·37-s − 0.468·41-s − 1.52·43-s + 0.875·47-s + 2/7·49-s − 0.686·53-s − 0.674·55-s − 0.520·59-s − 0.896·61-s − 0.124·65-s + 1.46·67-s − 0.593·71-s + 0.234·73-s − 1.70·77-s − 1.46·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.040300817419889487340992272551, −7.32350601773238389640751272254, −6.51636016703422468528885784305, −5.48927472426394427749267003240, −5.13124238045700232058062769052, −4.36431631260615805790249240018, −3.22898632671117123034544099556, −2.27512072123423792723142770388, −1.59234257892317960562059000064, 0,
1.59234257892317960562059000064, 2.27512072123423792723142770388, 3.22898632671117123034544099556, 4.36431631260615805790249240018, 5.13124238045700232058062769052, 5.48927472426394427749267003240, 6.51636016703422468528885784305, 7.32350601773238389640751272254, 8.040300817419889487340992272551