| L(s) = 1 | − 2.39·3-s − 1.13·7-s + 2.75·9-s − 2.01·11-s − 2.26·13-s + 1.08·17-s + 5.58·19-s + 2.72·21-s − 8.33·23-s + 0.591·27-s + 9.11·29-s − 8.75·31-s + 4.83·33-s + 2.54·37-s + 5.42·39-s + 9.82·41-s + 2.91·43-s + 2.09·47-s − 5.71·49-s − 2.59·51-s + 10.5·53-s − 13.3·57-s + 5.53·59-s − 1.63·61-s − 3.12·63-s + 10.5·67-s + 19.9·69-s + ⋯ |
| L(s) = 1 | − 1.38·3-s − 0.429·7-s + 0.917·9-s − 0.608·11-s − 0.627·13-s + 0.262·17-s + 1.28·19-s + 0.594·21-s − 1.73·23-s + 0.113·27-s + 1.69·29-s − 1.57·31-s + 0.842·33-s + 0.418·37-s + 0.869·39-s + 1.53·41-s + 0.444·43-s + 0.304·47-s − 0.815·49-s − 0.364·51-s + 1.44·53-s − 1.77·57-s + 0.720·59-s − 0.209·61-s − 0.393·63-s + 1.29·67-s + 2.40·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7287179921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7287179921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.39T + 3T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 - 9.11T + 29T^{2} \) |
| 31 | \( 1 + 8.75T + 31T^{2} \) |
| 37 | \( 1 - 2.54T + 37T^{2} \) |
| 41 | \( 1 - 9.82T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 5.53T + 59T^{2} \) |
| 61 | \( 1 + 1.63T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 6.84T + 79T^{2} \) |
| 83 | \( 1 - 2.81T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−10.05405447131922687042495649100, −9.485929206143727185686810689449, −8.133786011535845071559390579758, −7.33362621496411984232201860138, −6.41942504501568628932123233917, −5.61998745621760884168344747670, −5.02543349921269280279835990871, −3.85183112850723098709754305594, −2.47559031039543173516635473788, −0.69440023213056820669337109707,
0.69440023213056820669337109707, 2.47559031039543173516635473788, 3.85183112850723098709754305594, 5.02543349921269280279835990871, 5.61998745621760884168344747670, 6.41942504501568628932123233917, 7.33362621496411984232201860138, 8.133786011535845071559390579758, 9.485929206143727185686810689449, 10.05405447131922687042495649100
