| L(s) = 1 | − 3-s + 9-s + 13-s − 6·17-s + 19-s + 2·23-s − 27-s − 3·37-s − 39-s − 8·43-s + 2·47-s + 6·51-s − 2·53-s − 57-s + 10·59-s + 5·61-s + 3·67-s − 2·69-s − 12·71-s + 13·73-s + 11·79-s + 81-s + 2·83-s + 6·89-s + 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.277·13-s − 1.45·17-s + 0.229·19-s + 0.417·23-s − 0.192·27-s − 0.493·37-s − 0.160·39-s − 1.21·43-s + 0.291·47-s + 0.840·51-s − 0.274·53-s − 0.132·57-s + 1.30·59-s + 0.640·61-s + 0.366·67-s − 0.240·69-s − 1.42·71-s + 1.52·73-s + 1.23·79-s + 1/9·81-s + 0.219·83-s + 0.635·89-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.62071829776572, −14.94578309937478, −14.42970684390931, −13.68774824284368, −13.25586917936455, −12.88032921765075, −12.13459985819259, −11.64525405428072, −11.19105090638548, −10.64330481258715, −10.17496585270955, −9.445249377785392, −8.944583490632639, −8.373943883363397, −7.735230854434167, −6.921371472167972, −6.651723008147709, −6.005285931374416, −5.208782894816667, −4.853177585822167, −4.042811205915730, −3.485457208804505, −2.536979473880521, −1.862726945460653, −0.9398448229058478, 0,
0.9398448229058478, 1.862726945460653, 2.536979473880521, 3.485457208804505, 4.042811205915730, 4.853177585822167, 5.208782894816667, 6.005285931374416, 6.651723008147709, 6.921371472167972, 7.735230854434167, 8.373943883363397, 8.944583490632639, 9.445249377785392, 10.17496585270955, 10.64330481258715, 11.19105090638548, 11.64525405428072, 12.13459985819259, 12.88032921765075, 13.25586917936455, 13.68774824284368, 14.42970684390931, 14.94578309937478, 15.62071829776572
