L(s) = 1 | − 3·5-s − 7-s − 11-s − 13-s − 5·19-s + 6·23-s + 4·25-s + 3·29-s + 4·31-s + 3·35-s − 7·37-s + 12·41-s − 2·43-s + 3·47-s + 49-s − 6·53-s + 3·55-s + 3·59-s + 2·61-s + 3·65-s + 67-s + 12·71-s − 7·73-s + 77-s + 4·79-s − 6·89-s + 91-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 0.377·7-s − 0.301·11-s − 0.277·13-s − 1.14·19-s + 1.25·23-s + 4/5·25-s + 0.557·29-s + 0.718·31-s + 0.507·35-s − 1.15·37-s + 1.87·41-s − 0.304·43-s + 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.404·55-s + 0.390·59-s + 0.256·61-s + 0.372·65-s + 0.122·67-s + 1.42·71-s − 0.819·73-s + 0.113·77-s + 0.450·79-s − 0.635·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.74390180178843, −16.02028725075063, −15.67451194824048, −15.18271683546963, −14.62958899106597, −13.99851833634735, −13.19900408188817, −12.65192254492612, −12.26043720017795, −11.59667300865559, −10.94824175029229, −10.58313682041843, −9.767976933004637, −9.044040974557456, −8.428000185066308, −7.938064608061662, −7.209157287584483, −6.743452908745466, −5.964450815600656, −5.021787680126356, −4.450916464333543, −3.780911727544894, −3.047038071966626, −2.305971383356267, −0.9459250541200533, 0,
0.9459250541200533, 2.305971383356267, 3.047038071966626, 3.780911727544894, 4.450916464333543, 5.021787680126356, 5.964450815600656, 6.743452908745466, 7.209157287584483, 7.938064608061662, 8.428000185066308, 9.044040974557456, 9.767976933004637, 10.58313682041843, 10.94824175029229, 11.59667300865559, 12.26043720017795, 12.65192254492612, 13.19900408188817, 13.99851833634735, 14.62958899106597, 15.18271683546963, 15.67451194824048, 16.02028725075063, 16.74390180178843