L(s) = 1 | + 2·7-s + 11-s − 2·17-s − 8·19-s + 2·23-s + 6·29-s + 2·37-s − 2·41-s + 4·43-s + 6·47-s − 3·49-s − 8·53-s − 8·59-s − 4·61-s + 12·67-s − 10·71-s + 6·73-s + 2·77-s + 10·79-s + 4·83-s − 10·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 0.301·11-s − 0.485·17-s − 1.83·19-s + 0.417·23-s + 1.11·29-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 0.875·47-s − 3/7·49-s − 1.09·53-s − 1.04·59-s − 0.512·61-s + 1.46·67-s − 1.18·71-s + 0.702·73-s + 0.227·77-s + 1.12·79-s + 0.439·83-s − 1.05·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.97901733387397, −14.64873286512557, −13.97537789673557, −13.63040210134655, −12.91223963883979, −12.41426381324188, −12.04975543912594, −11.21709528401604, −10.88381816781520, −10.55357387520911, −9.701389430882492, −9.209563553574973, −8.568539019329361, −8.199356350481946, −7.659752142515327, −6.756880725951004, −6.535668566467769, −5.813463581502796, −5.075340962710734, −4.431999819817870, −4.171875617222700, −3.195406746649035, −2.460254967513808, −1.838852391906280, −1.056207043650373, 0,
1.056207043650373, 1.838852391906280, 2.460254967513808, 3.195406746649035, 4.171875617222700, 4.431999819817870, 5.075340962710734, 5.813463581502796, 6.535668566467769, 6.756880725951004, 7.659752142515327, 8.199356350481946, 8.568539019329361, 9.209563553574973, 9.701389430882492, 10.55357387520911, 10.88381816781520, 11.21709528401604, 12.04975543912594, 12.41426381324188, 12.91223963883979, 13.63040210134655, 13.97537789673557, 14.64873286512557, 14.97901733387397