| L(s) = 1 | − 2-s − 2.35·3-s + 4-s − 4.10·5-s + 2.35·6-s + 7-s − 8-s + 2.56·9-s + 4.10·10-s − 2.03·11-s − 2.35·12-s + 5.49·13-s − 14-s + 9.67·15-s + 16-s + 6.00·17-s − 2.56·18-s − 4.10·20-s − 2.35·21-s + 2.03·22-s + 1.22·23-s + 2.35·24-s + 11.8·25-s − 5.49·26-s + 1.01·27-s + 28-s + 3.38·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.36·3-s + 0.5·4-s − 1.83·5-s + 0.963·6-s + 0.377·7-s − 0.353·8-s + 0.856·9-s + 1.29·10-s − 0.613·11-s − 0.681·12-s + 1.52·13-s − 0.267·14-s + 2.49·15-s + 0.250·16-s + 1.45·17-s − 0.605·18-s − 0.916·20-s − 0.514·21-s + 0.433·22-s + 0.254·23-s + 0.481·24-s + 2.36·25-s − 1.07·26-s + 0.195·27-s + 0.188·28-s + 0.628·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5661066229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5661066229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 - 6.00T + 17T^{2} \) |
| 23 | \( 1 - 1.22T + 23T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 - 8.56T + 37T^{2} \) |
| 41 | \( 1 + 1.69T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 9.20T + 53T^{2} \) |
| 59 | \( 1 - 0.753T + 59T^{2} \) |
| 61 | \( 1 + 1.75T + 61T^{2} \) |
| 67 | \( 1 + 5.87T + 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 5.55T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 9.13T + 97T^{2} \) |
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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
−8.235159338206447984724990721465, −7.65342608616463839870757072813, −6.84015969477277549900571438269, −6.20189139513717076787381093437, −5.33071088419125319731575481367, −4.66988154981769237851729377274, −3.72240360890999629098586292951, −3.02387283139055381053749117111, −1.24031535032165674362879338410, −0.58013871594329329865771334900,
0.58013871594329329865771334900, 1.24031535032165674362879338410, 3.02387283139055381053749117111, 3.72240360890999629098586292951, 4.66988154981769237851729377274, 5.33071088419125319731575481367, 6.20189139513717076787381093437, 6.84015969477277549900571438269, 7.65342608616463839870757072813, 8.235159338206447984724990721465
