L(s) = 1 | − 0.254·2-s − 3-s − 1.93·4-s − 0.644·5-s + 0.254·6-s + 3.73·7-s + 1.00·8-s + 9-s + 0.164·10-s − 2.16·11-s + 1.93·12-s + 13-s − 0.952·14-s + 0.644·15-s + 3.61·16-s + 6.60·17-s − 0.254·18-s + 5.07·19-s + 1.24·20-s − 3.73·21-s + 0.550·22-s − 1.31·23-s − 1.00·24-s − 4.58·25-s − 0.254·26-s − 27-s − 7.23·28-s + ⋯ |
L(s) = 1 | − 0.180·2-s − 0.577·3-s − 0.967·4-s − 0.288·5-s + 0.103·6-s + 1.41·7-s + 0.354·8-s + 0.333·9-s + 0.0519·10-s − 0.651·11-s + 0.558·12-s + 0.277·13-s − 0.254·14-s + 0.166·15-s + 0.903·16-s + 1.60·17-s − 0.0600·18-s + 1.16·19-s + 0.278·20-s − 0.815·21-s + 0.117·22-s − 0.274·23-s − 0.204·24-s − 0.916·25-s − 0.0499·26-s − 0.192·27-s − 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.299422161\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.299422161\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 0.254T + 2T^{2} \) |
| 5 | \( 1 + 0.644T + 5T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 17 | \( 1 - 6.60T + 17T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 + 1.31T + 23T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 - 6.80T + 43T^{2} \) |
| 47 | \( 1 - 4.86T + 47T^{2} \) |
| 53 | \( 1 - 1.28T + 53T^{2} \) |
| 59 | \( 1 - 4.33T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 1.97T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 7.92T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.316536711759498253634303896449, −7.69519487867851420830037120033, −7.42111818479950308562104905158, −5.84123405198005916409645292689, −5.46633212530904906513567298575, −4.74429603708481587868438468271, −4.07935319799708698470733609102, −3.08885760363245295303783304197, −1.58247648289828497471160835952, −0.76318629579711004644147789725,
0.76318629579711004644147789725, 1.58247648289828497471160835952, 3.08885760363245295303783304197, 4.07935319799708698470733609102, 4.74429603708481587868438468271, 5.46633212530904906513567298575, 5.84123405198005916409645292689, 7.42111818479950308562104905158, 7.69519487867851420830037120033, 8.316536711759498253634303896449