L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s + 11-s + 2·15-s + 4·17-s − 6·19-s + 2·21-s − 25-s − 27-s − 8·29-s − 8·31-s − 33-s + 4·35-s − 10·37-s − 8·41-s + 2·43-s − 2·45-s − 8·47-s − 3·49-s − 4·51-s − 2·53-s − 2·55-s + 6·57-s + 12·59-s + 10·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.516·15-s + 0.970·17-s − 1.37·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.174·33-s + 0.676·35-s − 1.64·37-s − 1.24·41-s + 0.304·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s − 0.269·55-s + 0.794·57-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.25496466858539, −13.43578243790896, −13.01170753875220, −12.57815227298574, −12.22844281975795, −11.56711952164505, −11.30382652191058, −10.70038381909452, −10.20694311894262, −9.684294167689569, −9.209746791396241, −8.511284175067852, −8.103102288694718, −7.479797735605734, −6.902407113849555, −6.612537544988117, −5.916232717756071, −5.314927475946371, −4.937035031155925, −3.926771317051120, −3.694685653246139, −3.331625935898923, −2.146196650675746, −1.706679487599166, −0.5805066102372999, 0,
0.5805066102372999, 1.706679487599166, 2.146196650675746, 3.331625935898923, 3.694685653246139, 3.926771317051120, 4.937035031155925, 5.314927475946371, 5.916232717756071, 6.612537544988117, 6.902407113849555, 7.479797735605734, 8.103102288694718, 8.511284175067852, 9.209746791396241, 9.684294167689569, 10.20694311894262, 10.70038381909452, 11.30382652191058, 11.56711952164505, 12.22844281975795, 12.57815227298574, 13.01170753875220, 13.43578243790896, 14.25496466858539