L(s) = 1 | + 1.61·3-s + 0.618·7-s − 0.381·9-s + 11-s + 2.23·13-s + 4.85·17-s + 5.47·19-s + 1.00·21-s − 6.32·23-s − 5.47·27-s − 4.38·29-s + 4.23·31-s + 1.61·33-s + 1.76·37-s + 3.61·39-s + 7.94·41-s + 8.47·43-s + 4.70·47-s − 6.61·49-s + 7.85·51-s − 3.85·53-s + 8.85·57-s + 3.76·59-s + 7.09·61-s − 0.236·63-s − 10.2·69-s + 0.291·71-s + ⋯ |
L(s) = 1 | + 0.934·3-s + 0.233·7-s − 0.127·9-s + 0.301·11-s + 0.620·13-s + 1.17·17-s + 1.25·19-s + 0.218·21-s − 1.31·23-s − 1.05·27-s − 0.813·29-s + 0.760·31-s + 0.281·33-s + 0.289·37-s + 0.579·39-s + 1.24·41-s + 1.29·43-s + 0.686·47-s − 0.945·49-s + 1.09·51-s − 0.529·53-s + 1.17·57-s + 0.490·59-s + 0.907·61-s − 0.0297·63-s − 1.23·69-s + 0.0346·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.003805846\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.003805846\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 + 3.85T + 53T^{2} \) |
| 59 | \( 1 - 3.76T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 0.291T + 71T^{2} \) |
| 73 | \( 1 + 7.09T + 73T^{2} \) |
| 79 | \( 1 + 2.85T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 4.90T + 97T^{2} \) |
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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
−8.223781723426664677529877244988, −7.82705826082705872095818859504, −7.17245950083537498448982838077, −5.91791898199049717297881902059, −5.66830965927934443366927677891, −4.41456832998777927976791898121, −3.64362760135930205747406393503, −3.00263122579617756114614593177, −2.03923691631822355179010684562, −0.972186085326020188009954674591,
0.972186085326020188009954674591, 2.03923691631822355179010684562, 3.00263122579617756114614593177, 3.64362760135930205747406393503, 4.41456832998777927976791898121, 5.66830965927934443366927677891, 5.91791898199049717297881902059, 7.17245950083537498448982838077, 7.82705826082705872095818859504, 8.223781723426664677529877244988