L(s) = 1 | + 2-s − 2.30·3-s + 4-s − 2.30·6-s − 3.30·7-s + 8-s + 2.30·9-s − 2.60·11-s − 2.30·12-s + 2.30·13-s − 3.30·14-s + 16-s − 1.30·17-s + 2.30·18-s − 0.605·19-s + 7.60·21-s − 2.60·22-s + 6.90·23-s − 2.30·24-s + 2.30·26-s + 1.60·27-s − 3.30·28-s − 29-s + 6.69·31-s + 32-s + 6·33-s − 1.30·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.32·3-s + 0.5·4-s − 0.940·6-s − 1.24·7-s + 0.353·8-s + 0.767·9-s − 0.785·11-s − 0.664·12-s + 0.638·13-s − 0.882·14-s + 0.250·16-s − 0.315·17-s + 0.542·18-s − 0.138·19-s + 1.65·21-s − 0.555·22-s + 1.44·23-s − 0.470·24-s + 0.451·26-s + 0.308·27-s − 0.624·28-s − 0.185·29-s + 1.20·31-s + 0.176·32-s + 1.04·33-s − 0.223·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.244068599\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.244068599\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + 0.605T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 - 0.605T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 + 3.30T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 7.69T + 59T^{2} \) |
| 61 | \( 1 - 0.302T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 8.90T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−9.773359255959282419713671592324, −8.789299973345270238139613721786, −7.60756474588340953838616825152, −6.57737266187076090558691723860, −6.29983320124433661949044005697, −5.38822582805257587068382646197, −4.70583445228624245113898702358, −3.55104236715666951181760716791, −2.60181460507936355609571553743, −0.75173783562438084876241770075,
0.75173783562438084876241770075, 2.60181460507936355609571553743, 3.55104236715666951181760716791, 4.70583445228624245113898702358, 5.38822582805257587068382646197, 6.29983320124433661949044005697, 6.57737266187076090558691723860, 7.60756474588340953838616825152, 8.789299973345270238139613721786, 9.773359255959282419713671592324