| L(s) = 1 | − 3-s − 7-s + 9-s + 6·13-s − 2·17-s − 6·19-s + 21-s + 2·23-s − 27-s − 6·29-s − 2·31-s − 4·37-s − 6·39-s − 8·41-s + 4·43-s + 4·47-s + 49-s + 2·51-s + 6·53-s + 6·57-s + 4·59-s − 14·61-s − 63-s + 4·67-s − 2·69-s + 10·73-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.485·17-s − 1.37·19-s + 0.218·21-s + 0.417·23-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.657·37-s − 0.960·39-s − 1.24·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.794·57-s + 0.520·59-s − 1.79·61-s − 0.125·63-s + 0.488·67-s − 0.240·69-s + 1.17·73-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.561476370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.561476370\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.80939522180916, −12.42750329581242, −11.95025724161240, −11.24010389380099, −11.04393729804455, −10.65221017704701, −10.21522621228918, −9.607877941366134, −8.927599466288507, −8.787180244695051, −8.290222695150277, −7.544918109898656, −7.124653435992216, −6.563528312650946, −6.084596271952685, −5.924904182128036, −5.125999441108108, −4.709012902987230, −3.970740684113191, −3.664365127678088, −3.164449661612973, −2.124116003907148, −1.908122481882219, −1.013811302124503, −0.4000516457353224,
0.4000516457353224, 1.013811302124503, 1.908122481882219, 2.124116003907148, 3.164449661612973, 3.664365127678088, 3.970740684113191, 4.709012902987230, 5.125999441108108, 5.924904182128036, 6.084596271952685, 6.563528312650946, 7.124653435992216, 7.544918109898656, 8.290222695150277, 8.787180244695051, 8.927599466288507, 9.607877941366134, 10.21522621228918, 10.65221017704701, 11.04393729804455, 11.24010389380099, 11.95025724161240, 12.42750329581242, 12.80939522180916
