| L(s) = 1 | − 2·2-s − 7.07·3-s + 4·4-s − 19.7·5-s + 14.1·6-s − 8·8-s + 23.0·9-s + 39.5·10-s − 14·11-s − 28.2·12-s + 50.9·13-s + 140·15-s + 16·16-s + 1.41·17-s − 46.0·18-s − 1.41·19-s − 79.1·20-s + 28·22-s + 140·23-s + 56.5·24-s + 267·25-s − 101.·26-s + 28.2·27-s − 286·29-s − 280·30-s − 93.3·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.36·3-s + 0.5·4-s − 1.77·5-s + 0.962·6-s − 0.353·8-s + 0.851·9-s + 1.25·10-s − 0.383·11-s − 0.680·12-s + 1.08·13-s + 2.40·15-s + 0.250·16-s + 0.0201·17-s − 0.602·18-s − 0.0170·19-s − 0.885·20-s + 0.271·22-s + 1.26·23-s + 0.481·24-s + 2.13·25-s − 0.768·26-s + 0.201·27-s − 1.83·29-s − 1.70·30-s − 0.540·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3748434021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3748434021\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 7.07T + 27T^{2} \) |
| 5 | \( 1 + 19.7T + 125T^{2} \) |
| 11 | \( 1 + 14T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.41T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.41T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140T + 1.21e4T^{2} \) |
| 29 | \( 1 + 286T + 2.43e4T^{2} \) |
| 31 | \( 1 + 93.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 38T + 5.06e4T^{2} \) |
| 41 | \( 1 + 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 34T + 7.95e4T^{2} \) |
| 47 | \( 1 - 523.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 74T + 1.48e5T^{2} \) |
| 59 | \( 1 - 434.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 14.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 684T + 3.00e5T^{2} \) |
| 71 | \( 1 - 588T + 3.57e5T^{2} \) |
| 73 | \( 1 + 270.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 422.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 618.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−12.95967567290602917403137230869, −11.96728387137672545605707381280, −11.14181711493088547971662539773, −10.76558519542085117149843533826, −8.924645867876475589411123580714, −7.76394760220724446805700400063, −6.77108089612175459885657561003, −5.30684781566433497682552596931, −3.71050791864670579925600787107, −0.64076081636971140382714899099,
0.64076081636971140382714899099, 3.71050791864670579925600787107, 5.30684781566433497682552596931, 6.77108089612175459885657561003, 7.76394760220724446805700400063, 8.924645867876475589411123580714, 10.76558519542085117149843533826, 11.14181711493088547971662539773, 11.96728387137672545605707381280, 12.95967567290602917403137230869
