| L(s) = 1 | + 2-s + 3.27·3-s + 4-s − 3.29·5-s + 3.27·6-s + 1.59·7-s + 8-s + 7.71·9-s − 3.29·10-s − 0.552·11-s + 3.27·12-s + 4.95·13-s + 1.59·14-s − 10.7·15-s + 16-s − 6.15·17-s + 7.71·18-s + 3.34·19-s − 3.29·20-s + 5.22·21-s − 0.552·22-s + 23-s + 3.27·24-s + 5.87·25-s + 4.95·26-s + 15.4·27-s + 1.59·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.88·3-s + 0.5·4-s − 1.47·5-s + 1.33·6-s + 0.603·7-s + 0.353·8-s + 2.57·9-s − 1.04·10-s − 0.166·11-s + 0.944·12-s + 1.37·13-s + 0.426·14-s − 2.78·15-s + 0.250·16-s − 1.49·17-s + 1.81·18-s + 0.767·19-s − 0.737·20-s + 1.14·21-s − 0.117·22-s + 0.208·23-s + 0.668·24-s + 1.17·25-s + 0.972·26-s + 2.97·27-s + 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.316027942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.316027942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.27T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 + 0.552T + 11T^{2} \) |
| 13 | \( 1 - 4.95T + 13T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + 1.25T + 37T^{2} \) |
| 41 | \( 1 + 8.64T + 41T^{2} \) |
| 43 | \( 1 - 9.52T + 43T^{2} \) |
| 47 | \( 1 + 0.00901T + 47T^{2} \) |
| 53 | \( 1 + 0.871T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 3.61T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 3.40T + 97T^{2} \) |
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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
−9.236565215141988577785452142023, −8.634120665946146114660532595520, −8.014501030076100943977894232338, −7.41370779236610188148252008400, −6.63982828759453607079650617286, −5.00620388713405612667326531863, −4.03491871523372297116246593067, −3.69629988794887347341854464120, −2.73747588511298536411500795183, −1.54725016103956798415865633780,
1.54725016103956798415865633780, 2.73747588511298536411500795183, 3.69629988794887347341854464120, 4.03491871523372297116246593067, 5.00620388713405612667326531863, 6.63982828759453607079650617286, 7.41370779236610188148252008400, 8.014501030076100943977894232338, 8.634120665946146114660532595520, 9.236565215141988577785452142023
