L(s) = 1 | + 2.59·2-s + 3-s + 4.75·4-s − 0.176·5-s + 2.59·6-s − 7-s + 7.17·8-s + 9-s − 0.458·10-s + 2.62·11-s + 4.75·12-s + 0.592·13-s − 2.59·14-s − 0.176·15-s + 9.12·16-s + 5.67·17-s + 2.59·18-s − 1.18·19-s − 0.839·20-s − 21-s + 6.81·22-s + 2.39·23-s + 7.17·24-s − 4.96·25-s + 1.54·26-s + 27-s − 4.75·28-s + ⋯ |
L(s) = 1 | + 1.83·2-s + 0.577·3-s + 2.37·4-s − 0.0788·5-s + 1.06·6-s − 0.377·7-s + 2.53·8-s + 0.333·9-s − 0.144·10-s + 0.790·11-s + 1.37·12-s + 0.164·13-s − 0.694·14-s − 0.0455·15-s + 2.28·16-s + 1.37·17-s + 0.612·18-s − 0.271·19-s − 0.187·20-s − 0.218·21-s + 1.45·22-s + 0.499·23-s + 1.46·24-s − 0.993·25-s + 0.302·26-s + 0.192·27-s − 0.899·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.344615986\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.344615986\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 5 | \( 1 + 0.176T + 5T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 - 0.592T + 13T^{2} \) |
| 17 | \( 1 - 5.67T + 17T^{2} \) |
| 19 | \( 1 + 1.18T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 + 5.27T + 29T^{2} \) |
| 31 | \( 1 - 1.78T + 31T^{2} \) |
| 37 | \( 1 - 9.35T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 47 | \( 1 - 3.36T + 47T^{2} \) |
| 53 | \( 1 + 9.83T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 + 0.198T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 3.87T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
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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
−7.70244752506257396601127358444, −6.85711481501051686375327086242, −6.31633950002964734617111163288, −5.66635034206703724088619678007, −4.97555806756136355388778473831, −4.03219964671449232247539200952, −3.70928316804277066618579800340, −2.98701726202917970841520335975, −2.18960269812056147253723349958, −1.21758798128124256845059320676,
1.21758798128124256845059320676, 2.18960269812056147253723349958, 2.98701726202917970841520335975, 3.70928316804277066618579800340, 4.03219964671449232247539200952, 4.97555806756136355388778473831, 5.66635034206703724088619678007, 6.31633950002964734617111163288, 6.85711481501051686375327086242, 7.70244752506257396601127358444