| L(s) = 1 | + 2-s + 3.41·3-s + 4-s − 1.27·5-s + 3.41·6-s − 7-s + 8-s + 8.63·9-s − 1.27·10-s + 2.74·11-s + 3.41·12-s − 1.37·13-s − 14-s − 4.33·15-s + 16-s − 4.10·17-s + 8.63·18-s + 5.71·19-s − 1.27·20-s − 3.41·21-s + 2.74·22-s − 3.98·23-s + 3.41·24-s − 3.38·25-s − 1.37·26-s + 19.2·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.96·3-s + 0.5·4-s − 0.568·5-s + 1.39·6-s − 0.377·7-s + 0.353·8-s + 2.87·9-s − 0.401·10-s + 0.827·11-s + 0.984·12-s − 0.380·13-s − 0.267·14-s − 1.11·15-s + 0.250·16-s − 0.995·17-s + 2.03·18-s + 1.31·19-s − 0.284·20-s − 0.744·21-s + 0.585·22-s − 0.831·23-s + 0.696·24-s − 0.677·25-s − 0.268·26-s + 3.70·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.543337134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.543337134\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 3.41T + 3T^{2} \) |
| 5 | \( 1 + 1.27T + 5T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 - 5.90T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 4.04T + 53T^{2} \) |
| 59 | \( 1 - 7.41T + 59T^{2} \) |
| 61 | \( 1 + 1.50T + 61T^{2} \) |
| 67 | \( 1 + 9.73T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 1.98T + 97T^{2} \) |
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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
−7.935653288843871908610414475448, −7.55047381488261717088228254565, −6.77701377235180467059117043302, −6.15458756097761112742806540222, −4.70374506960005656547346925513, −4.26751527555509106522298264757, −3.58139300531623195732510848779, −2.91687538370603132087127395700, −2.25852773147430769014623553055, −1.19588118956044200056140096818,
1.19588118956044200056140096818, 2.25852773147430769014623553055, 2.91687538370603132087127395700, 3.58139300531623195732510848779, 4.26751527555509106522298264757, 4.70374506960005656547346925513, 6.15458756097761112742806540222, 6.77701377235180467059117043302, 7.55047381488261717088228254565, 7.935653288843871908610414475448
