L(s) = 1 | + 5-s − 4·7-s − 11-s + 2·13-s − 2·17-s + 4·19-s + 4·23-s + 25-s + 10·29-s − 8·31-s − 4·35-s − 2·37-s − 10·41-s − 12·43-s − 12·47-s + 9·49-s + 6·53-s − 55-s + 12·59-s + 10·61-s + 2·65-s − 4·67-s + 12·71-s − 14·73-s + 4·77-s − 12·79-s + 12·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.85·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s − 1.56·41-s − 1.82·43-s − 1.75·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s + 1.56·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s + 1.42·71-s − 1.63·73-s + 0.455·77-s − 1.35·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.22962516013465535614154462900, −6.74816447540167279541498870891, −6.28419099610732097086551323073, −5.37154794256349511353534106093, −4.86847941182743977494584828586, −3.56615521604607497481147659445, −3.26842014808120251463369498108, −2.34280989767668746206154786581, −1.21269081161787424197241569425, 0,
1.21269081161787424197241569425, 2.34280989767668746206154786581, 3.26842014808120251463369498108, 3.56615521604607497481147659445, 4.86847941182743977494584828586, 5.37154794256349511353534106093, 6.28419099610732097086551323073, 6.74816447540167279541498870891, 7.22962516013465535614154462900