L(s) = 1 | + 2·3-s − 3·5-s − 7-s + 9-s + 13-s − 6·15-s − 6·17-s + 7·19-s − 2·21-s − 3·23-s + 4·25-s − 4·27-s − 9·29-s − 5·31-s + 3·35-s + 2·37-s + 2·39-s − 6·41-s + 43-s − 3·45-s − 3·47-s + 49-s − 12·51-s − 9·53-s + 14·57-s − 10·61-s − 63-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.277·13-s − 1.54·15-s − 1.45·17-s + 1.60·19-s − 0.436·21-s − 0.625·23-s + 4/5·25-s − 0.769·27-s − 1.67·29-s − 0.898·31-s + 0.507·35-s + 0.328·37-s + 0.320·39-s − 0.937·41-s + 0.152·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s − 1.68·51-s − 1.23·53-s + 1.85·57-s − 1.28·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−9.148001222079290166548692208845, −8.222490883796533571515292244064, −7.67278082338318419108930940202, −7.02868293890643359091621042595, −5.84786552632357662072719983468, −4.61390554680519199978963802626, −3.64885443653876343577018155811, −3.23023689233311204016664652811, −1.94321857646898970494164375533, 0,
1.94321857646898970494164375533, 3.23023689233311204016664652811, 3.64885443653876343577018155811, 4.61390554680519199978963802626, 5.84786552632357662072719983468, 7.02868293890643359091621042595, 7.67278082338318419108930940202, 8.222490883796533571515292244064, 9.148001222079290166548692208845