L(s) = 1 | − 2·3-s − 2·4-s − 3·5-s + 7-s + 9-s + 4·12-s + 13-s + 6·15-s + 4·16-s − 6·17-s − 7·19-s + 6·20-s − 2·21-s + 3·23-s + 4·25-s + 4·27-s − 2·28-s − 9·29-s + 5·31-s − 3·35-s − 2·36-s + 2·37-s − 2·39-s − 6·41-s − 43-s − 3·45-s + 3·47-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.15·12-s + 0.277·13-s + 1.54·15-s + 16-s − 1.45·17-s − 1.60·19-s + 1.34·20-s − 0.436·21-s + 0.625·23-s + 4/5·25-s + 0.769·27-s − 0.377·28-s − 1.67·29-s + 0.898·31-s − 0.507·35-s − 1/3·36-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 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
−13.32859477024006795390919405039, −12.40148043503926800726298674034, −11.34343268397352416353725012574, −10.73267410274711374905403265951, −8.965474064429713311069674592028, −8.023034081311283559976527256533, −6.48432181419363077598714474936, −4.95454046107923865998450494116, −4.04609643315305459582542236808, 0,
4.04609643315305459582542236808, 4.95454046107923865998450494116, 6.48432181419363077598714474936, 8.023034081311283559976527256533, 8.965474064429713311069674592028, 10.73267410274711374905403265951, 11.34343268397352416353725012574, 12.40148043503926800726298674034, 13.32859477024006795390919405039