L(s) = 1 | − 5.41·2-s + 4.17·3-s + 21.3·4-s − 5·5-s − 22.6·6-s − 25.0·7-s − 72.4·8-s − 9.58·9-s + 27.0·10-s − 11·11-s + 89.2·12-s − 82.0·13-s + 135.·14-s − 20.8·15-s + 221.·16-s − 5.77·17-s + 51.9·18-s − 19·19-s − 106.·20-s − 104.·21-s + 59.6·22-s − 202.·23-s − 302.·24-s + 25·25-s + 444.·26-s − 152.·27-s − 535.·28-s + ⋯ |
L(s) = 1 | − 1.91·2-s + 0.803·3-s + 2.67·4-s − 0.447·5-s − 1.53·6-s − 1.35·7-s − 3.20·8-s − 0.354·9-s + 0.856·10-s − 0.301·11-s + 2.14·12-s − 1.75·13-s + 2.59·14-s − 0.359·15-s + 3.46·16-s − 0.0824·17-s + 0.679·18-s − 0.229·19-s − 1.19·20-s − 1.08·21-s + 0.577·22-s − 1.83·23-s − 2.57·24-s + 0.200·25-s + 3.35·26-s − 1.08·27-s − 3.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.04973598025\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04973598025\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
good | 2 | \( 1 + 5.41T + 8T^{2} \) |
| 3 | \( 1 - 4.17T + 27T^{2} \) |
| 7 | \( 1 + 25.0T + 343T^{2} \) |
| 13 | \( 1 + 82.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.77T + 4.91e3T^{2} \) |
| 23 | \( 1 + 202.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 9.64T + 2.97e4T^{2} \) |
| 37 | \( 1 + 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 106.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 268.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 379.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 566.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 202.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 360.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 523.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 454.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 744.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 210.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 908.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 80.6T + 9.12e5T^{2} \) |
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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
−9.434802194143266696523683847295, −8.857044050487496739791557842325, −7.954872581171602690978714375026, −7.44619829995707091983588992540, −6.65326430920844384245108354633, −5.65039934229236951951045457491, −3.69820653189711274435205980365, −2.72650465252038978724597397151, −2.08320642033636573536993723615, −0.13633482657469637378071836534,
0.13633482657469637378071836534, 2.08320642033636573536993723615, 2.72650465252038978724597397151, 3.69820653189711274435205980365, 5.65039934229236951951045457491, 6.65326430920844384245108354633, 7.44619829995707091983588992540, 7.954872581171602690978714375026, 8.857044050487496739791557842325, 9.434802194143266696523683847295