| L(s) = 1 | − 2.26·2-s + 0.654·3-s + 3.14·4-s − 0.968·5-s − 1.48·6-s + 1.81·7-s − 2.60·8-s − 2.57·9-s + 2.19·10-s + 11-s + 2.05·12-s + 3.55·13-s − 4.10·14-s − 0.633·15-s − 0.386·16-s + 17-s + 5.83·18-s − 4.84·19-s − 3.04·20-s + 1.18·21-s − 2.26·22-s − 7.36·23-s − 1.70·24-s − 4.06·25-s − 8.07·26-s − 3.64·27-s + 5.69·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 0.377·3-s + 1.57·4-s − 0.432·5-s − 0.606·6-s + 0.684·7-s − 0.920·8-s − 0.857·9-s + 0.694·10-s + 0.301·11-s + 0.594·12-s + 0.986·13-s − 1.09·14-s − 0.163·15-s − 0.0967·16-s + 0.242·17-s + 1.37·18-s − 1.11·19-s − 0.681·20-s + 0.258·21-s − 0.483·22-s − 1.53·23-s − 0.347·24-s − 0.812·25-s − 1.58·26-s − 0.701·27-s + 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6953834816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6953834816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 - 0.654T + 3T^{2} \) |
| 5 | \( 1 + 0.968T + 5T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 + 7.36T + 23T^{2} \) |
| 29 | \( 1 - 7.99T + 29T^{2} \) |
| 31 | \( 1 + 2.59T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 1.85T + 41T^{2} \) |
| 47 | \( 1 + 3.71T + 47T^{2} \) |
| 53 | \( 1 + 8.29T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 4.30T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 2.83T + 73T^{2} \) |
| 79 | \( 1 - 0.315T + 79T^{2} \) |
| 83 | \( 1 - 9.58T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.111616288254653504064803913640, −7.61810807343855566299762491108, −6.50767374995040105448355564592, −6.19059611307542575146989022400, −5.07585177331835781988530857595, −4.09481397998763971987068605933, −3.38724189441597482968073824018, −2.22340301493473125798366361630, −1.69194034008902729328184566829, −0.51404590032146868464167953986,
0.51404590032146868464167953986, 1.69194034008902729328184566829, 2.22340301493473125798366361630, 3.38724189441597482968073824018, 4.09481397998763971987068605933, 5.07585177331835781988530857595, 6.19059611307542575146989022400, 6.50767374995040105448355564592, 7.61810807343855566299762491108, 8.111616288254653504064803913640
