| L(s) = 1 | − 5-s − 3·7-s + 2·11-s − 2·13-s − 5·17-s + 2·19-s − 23-s + 25-s + 5·29-s − 3·31-s + 3·35-s − 7·37-s + 11·41-s + 8·43-s + 8·47-s + 2·49-s − 5·53-s − 2·55-s − 59-s − 8·61-s + 2·65-s + 9·67-s + 71-s + 10·73-s − 6·77-s + 15·83-s + 5·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 0.603·11-s − 0.554·13-s − 1.21·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 0.928·29-s − 0.538·31-s + 0.507·35-s − 1.15·37-s + 1.71·41-s + 1.21·43-s + 1.16·47-s + 2/7·49-s − 0.686·53-s − 0.269·55-s − 0.130·59-s − 1.02·61-s + 0.248·65-s + 1.09·67-s + 0.118·71-s + 1.17·73-s − 0.683·77-s + 1.64·83-s + 0.542·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.09983174543628, −15.64954492106642, −15.32424441752567, −14.43309896989164, −14.01837555593489, −13.47630016830013, −12.64400268923747, −12.42834321405877, −11.87584860075330, −11.01623566927185, −10.73356943193613, −9.891858966256347, −9.235418562112920, −9.083400856671502, −8.155702453052199, −7.489497617571587, −6.875608254906773, −6.428739087676451, −5.753439233055493, −4.895890123578519, −4.190139872444525, −3.642932124678263, −2.834591828424408, −2.179930143093416, −0.9417841583936170, 0,
0.9417841583936170, 2.179930143093416, 2.834591828424408, 3.642932124678263, 4.190139872444525, 4.895890123578519, 5.753439233055493, 6.428739087676451, 6.875608254906773, 7.489497617571587, 8.155702453052199, 9.083400856671502, 9.235418562112920, 9.891858966256347, 10.73356943193613, 11.01623566927185, 11.87584860075330, 12.42834321405877, 12.64400268923747, 13.47630016830013, 14.01837555593489, 14.43309896989164, 15.32424441752567, 15.64954492106642, 16.09983174543628
