| L(s) = 1 | + 2-s − 1.57·3-s + 4-s − 0.555·5-s − 1.57·6-s − 2.43·7-s + 8-s − 0.527·9-s − 0.555·10-s − 11-s − 1.57·12-s − 6.97·13-s − 2.43·14-s + 0.874·15-s + 16-s − 1.73·17-s − 0.527·18-s − 1.59·19-s − 0.555·20-s + 3.82·21-s − 22-s − 5.45·23-s − 1.57·24-s − 4.69·25-s − 6.97·26-s + 5.54·27-s − 2.43·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.907·3-s + 0.5·4-s − 0.248·5-s − 0.641·6-s − 0.920·7-s + 0.353·8-s − 0.175·9-s − 0.175·10-s − 0.301·11-s − 0.453·12-s − 1.93·13-s − 0.650·14-s + 0.225·15-s + 0.250·16-s − 0.421·17-s − 0.124·18-s − 0.366·19-s − 0.124·20-s + 0.835·21-s − 0.213·22-s − 1.13·23-s − 0.320·24-s − 0.938·25-s − 1.36·26-s + 1.06·27-s − 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8381240735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8381240735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 1.57T + 3T^{2} \) |
| 5 | \( 1 + 0.555T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 13 | \( 1 + 6.97T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 + 1.59T + 19T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 - 7.42T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 - 1.28T + 37T^{2} \) |
| 41 | \( 1 - 8.56T + 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 - 7.77T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 5.22T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 1.14T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−8.110427685118865084154507209591, −7.51565532772258672486205023580, −6.56703145232785567831376940341, −6.23697862513627369093818673635, −5.38002025064362964061133189117, −4.71643981291078281449618915633, −4.02110376087758919759392769642, −2.85568064389659597956773314228, −2.29764428576539347600606048038, −0.44884376028293003658599651402,
0.44884376028293003658599651402, 2.29764428576539347600606048038, 2.85568064389659597956773314228, 4.02110376087758919759392769642, 4.71643981291078281449618915633, 5.38002025064362964061133189117, 6.23697862513627369093818673635, 6.56703145232785567831376940341, 7.51565532772258672486205023580, 8.110427685118865084154507209591
