| L(s) = 1 | − 3-s + 3·7-s + 9-s + 5·11-s + 3·13-s + 6·17-s − 19-s − 3·21-s − 23-s − 27-s + 5·29-s − 4·31-s − 5·33-s − 8·37-s − 3·39-s + 9·41-s + 9·43-s + 2·47-s + 2·49-s − 6·51-s + 12·53-s + 57-s + 6·59-s − 8·61-s + 3·63-s − 4·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.832·13-s + 1.45·17-s − 0.229·19-s − 0.654·21-s − 0.208·23-s − 0.192·27-s + 0.928·29-s − 0.718·31-s − 0.870·33-s − 1.31·37-s − 0.480·39-s + 1.40·41-s + 1.37·43-s + 0.291·47-s + 2/7·49-s − 0.840·51-s + 1.64·53-s + 0.132·57-s + 0.781·59-s − 1.02·61-s + 0.377·63-s − 0.488·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.843508149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.843508149\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.94845309126350, −13.15803344559207, −12.32439080810730, −12.25437321191431, −11.74458472952054, −11.23618689114105, −10.77245093101402, −10.43234757049512, −9.703262378836291, −9.239817224685357, −8.616392114444037, −8.334128330188621, −7.531398141105996, −7.250335173554910, −6.550460625922034, −5.978442808548936, −5.620828821676076, −5.044913915154162, −4.318378116353835, −3.960107989992141, −3.426340301707860, −2.512823711104460, −1.676893878446701, −1.223227126150538, −0.7251806433632047,
0.7251806433632047, 1.223227126150538, 1.676893878446701, 2.512823711104460, 3.426340301707860, 3.960107989992141, 4.318378116353835, 5.044913915154162, 5.620828821676076, 5.978442808548936, 6.550460625922034, 7.250335173554910, 7.531398141105996, 8.334128330188621, 8.616392114444037, 9.239817224685357, 9.703262378836291, 10.43234757049512, 10.77245093101402, 11.23618689114105, 11.74458472952054, 12.25437321191431, 12.32439080810730, 13.15803344559207, 13.94845309126350
