L(s) = 1 | − 0.470·2-s − 1.77·4-s + 0.529·5-s + 1.77·8-s − 0.249·10-s + 2.24·11-s − 13-s + 2.71·16-s − 1.30·17-s + 1.47·19-s − 0.941·20-s − 1.05·22-s − 5.83·23-s − 4.71·25-s + 0.470·26-s − 5.22·29-s + 7.02·31-s − 4.83·32-s + 0.615·34-s − 2.36·37-s − 0.692·38-s + 0.941·40-s + 6.49·41-s + 11.3·43-s − 4.00·44-s + 2.74·46-s + 8.58·47-s + ⋯ |
L(s) = 1 | − 0.332·2-s − 0.889·4-s + 0.236·5-s + 0.628·8-s − 0.0787·10-s + 0.678·11-s − 0.277·13-s + 0.679·16-s − 0.317·17-s + 0.337·19-s − 0.210·20-s − 0.225·22-s − 1.21·23-s − 0.943·25-s + 0.0923·26-s − 0.969·29-s + 1.26·31-s − 0.855·32-s + 0.105·34-s − 0.389·37-s − 0.112·38-s + 0.148·40-s + 1.01·41-s + 1.73·43-s − 0.603·44-s + 0.405·46-s + 1.25·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + 0.470T + 2T^{2} \) |
| 5 | \( 1 - 0.529T + 5T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 - 1.47T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 + 5.22T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 + 7.64T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 6.91T + 89T^{2} \) |
| 97 | \( 1 - 3.47T + 97T^{2} \) |
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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
−7.66325094295565002468327797602, −7.41490125714622245671848529832, −6.00173115838023391012777872041, −5.90859225246227753217270002182, −4.61848616347178031920120262817, −4.25383830594107742839147924885, −3.35243017741884140470896179548, −2.18648840239592804521063851983, −1.21638248487755075255628957140, 0,
1.21638248487755075255628957140, 2.18648840239592804521063851983, 3.35243017741884140470896179548, 4.25383830594107742839147924885, 4.61848616347178031920120262817, 5.90859225246227753217270002182, 6.00173115838023391012777872041, 7.41490125714622245671848529832, 7.66325094295565002468327797602