L(s) = 1 | + 5-s + 6·11-s − 6·17-s + 4·19-s − 6·23-s + 25-s + 2·29-s − 8·31-s − 2·37-s − 10·41-s − 12·43-s − 8·47-s + 2·53-s + 6·55-s − 4·59-s − 8·61-s − 16·67-s + 10·71-s + 4·79-s + 4·83-s − 6·85-s + 6·89-s + 4·95-s − 8·97-s + 10·101-s − 12·103-s + 2·107-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.80·11-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.328·37-s − 1.56·41-s − 1.82·43-s − 1.16·47-s + 0.274·53-s + 0.809·55-s − 0.520·59-s − 1.02·61-s − 1.95·67-s + 1.18·71-s + 0.450·79-s + 0.439·83-s − 0.650·85-s + 0.635·89-s + 0.410·95-s − 0.812·97-s + 0.995·101-s − 1.18·103-s + 0.193·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.25610444253577179409503258094, −6.57934485339180987473185462837, −6.28524850725879777493617066587, −5.31786670221020771240676352839, −4.61120227572589837330919579502, −3.81449757633236707852268658048, −3.20079397928451625204775483210, −1.92268851363993773520446399268, −1.50449592100046886633648806228, 0,
1.50449592100046886633648806228, 1.92268851363993773520446399268, 3.20079397928451625204775483210, 3.81449757633236707852268658048, 4.61120227572589837330919579502, 5.31786670221020771240676352839, 6.28524850725879777493617066587, 6.57934485339180987473185462837, 7.25610444253577179409503258094