L(s) = 1 | + 5-s − 7-s + 11-s + 2·13-s + 5·17-s + 7·19-s + 6·23-s + 25-s − 29-s − 5·31-s − 35-s − 11·37-s − 2·41-s − 4·43-s − 6·47-s − 6·49-s − 53-s + 55-s − 10·59-s − 5·61-s + 2·65-s + 8·67-s − 9·71-s − 6·73-s − 77-s − 10·79-s − 6·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.301·11-s + 0.554·13-s + 1.21·17-s + 1.60·19-s + 1.25·23-s + 1/5·25-s − 0.185·29-s − 0.898·31-s − 0.169·35-s − 1.80·37-s − 0.312·41-s − 0.609·43-s − 0.875·47-s − 6/7·49-s − 0.137·53-s + 0.134·55-s − 1.30·59-s − 0.640·61-s + 0.248·65-s + 0.977·67-s − 1.06·71-s − 0.702·73-s − 0.113·77-s − 1.12·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−15.48436439909503, −14.61369926368823, −14.31183815424475, −13.78431721665155, −13.21535870150880, −12.77039271031397, −12.16002375114736, −11.62653253903740, −11.11372474794752, −10.46193615371278, −9.869223192958404, −9.526011692665329, −8.862421109181866, −8.424769140117076, −7.449705447246313, −7.264005171118320, −6.460517619202224, −5.904605513738320, −5.214452899713103, −4.937516286737482, −3.756821631244974, −3.296072641046842, −2.853322164727887, −1.503504096663033, −1.332398872261024, 0,
1.332398872261024, 1.503504096663033, 2.853322164727887, 3.296072641046842, 3.756821631244974, 4.937516286737482, 5.214452899713103, 5.904605513738320, 6.460517619202224, 7.264005171118320, 7.449705447246313, 8.424769140117076, 8.862421109181866, 9.526011692665329, 9.869223192958404, 10.46193615371278, 11.11372474794752, 11.62653253903740, 12.16002375114736, 12.77039271031397, 13.21535870150880, 13.78431721665155, 14.31183815424475, 14.61369926368823, 15.48436439909503