| L(s) = 1 | − 3-s + 9-s − 11-s − 8·17-s − 2·19-s + 6·23-s − 27-s + 6·29-s + 6·31-s + 33-s + 2·37-s + 8·41-s − 7·43-s + 7·47-s + 8·51-s − 7·53-s + 2·57-s + 10·59-s − 2·61-s − 16·67-s − 6·69-s − 12·71-s + 7·73-s + 81-s + 9·83-s − 6·87-s − 89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.94·17-s − 0.458·19-s + 1.25·23-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.174·33-s + 0.328·37-s + 1.24·41-s − 1.06·43-s + 1.02·47-s + 1.12·51-s − 0.961·53-s + 0.264·57-s + 1.30·59-s − 0.256·61-s − 1.95·67-s − 0.722·69-s − 1.42·71-s + 0.819·73-s + 1/9·81-s + 0.987·83-s − 0.643·87-s − 0.105·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.90883333898777, −12.35452688432540, −11.91226024186677, −11.45012510387322, −10.97468167866254, −10.65827407659370, −10.30080876459183, −9.628025139633006, −9.183991967611215, −8.717483402841863, −8.330557973756307, −7.746323691564198, −7.148162284542884, −6.712901910013232, −6.385676478427295, −5.902425332260677, −5.228101139267645, −4.755950536272133, −4.369697566479541, −3.990653857075564, −2.963638057455922, −2.726423680644452, −2.069904941244743, −1.335621483250924, −0.6960994823194855, 0,
0.6960994823194855, 1.335621483250924, 2.069904941244743, 2.726423680644452, 2.963638057455922, 3.990653857075564, 4.369697566479541, 4.755950536272133, 5.228101139267645, 5.902425332260677, 6.385676478427295, 6.712901910013232, 7.148162284542884, 7.746323691564198, 8.330557973756307, 8.717483402841863, 9.183991967611215, 9.628025139633006, 10.30080876459183, 10.65827407659370, 10.97468167866254, 11.45012510387322, 11.91226024186677, 12.35452688432540, 12.90883333898777
