L(s) = 1 | + 5-s + 7-s + 3·11-s + 13-s − 3·17-s − 2·19-s + 3·23-s + 25-s − 6·29-s − 2·31-s + 35-s − 7·37-s − 9·41-s − 8·43-s − 6·47-s − 6·49-s − 3·53-s + 3·55-s − 7·61-s + 65-s + 4·67-s + 3·71-s − 10·73-s + 3·77-s + 79-s − 3·85-s − 3·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.904·11-s + 0.277·13-s − 0.727·17-s − 0.458·19-s + 0.625·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.169·35-s − 1.15·37-s − 1.40·41-s − 1.21·43-s − 0.875·47-s − 6/7·49-s − 0.412·53-s + 0.404·55-s − 0.896·61-s + 0.124·65-s + 0.488·67-s + 0.356·71-s − 1.17·73-s + 0.341·77-s + 0.112·79-s − 0.325·85-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−7.19134525440628780922985605234, −6.69311559182299210558445082684, −6.10515413782953139301523040081, −5.21991018037274225094673667739, −4.68788144157411426097343216983, −3.77841506338740512174952624718, −3.13495638728606196402916962038, −1.91443139172699666471871571259, −1.50167981080022529632026249416, 0,
1.50167981080022529632026249416, 1.91443139172699666471871571259, 3.13495638728606196402916962038, 3.77841506338740512174952624718, 4.68788144157411426097343216983, 5.21991018037274225094673667739, 6.10515413782953139301523040081, 6.69311559182299210558445082684, 7.19134525440628780922985605234