| L(s) = 1 | − 2.98·3-s − 0.559·5-s − 7-s + 5.93·9-s − 11-s − 13-s + 1.67·15-s − 4.23·17-s + 2.00·19-s + 2.98·21-s + 6.52·23-s − 4.68·25-s − 8.76·27-s − 2.70·29-s − 2.02·31-s + 2.98·33-s + 0.559·35-s + 0.402·37-s + 2.98·39-s − 7.50·41-s − 0.488·43-s − 3.31·45-s + 1.94·47-s + 49-s + 12.6·51-s + 7.42·53-s + 0.559·55-s + ⋯ |
| L(s) = 1 | − 1.72·3-s − 0.250·5-s − 0.377·7-s + 1.97·9-s − 0.301·11-s − 0.277·13-s + 0.431·15-s − 1.02·17-s + 0.459·19-s + 0.652·21-s + 1.35·23-s − 0.937·25-s − 1.68·27-s − 0.502·29-s − 0.363·31-s + 0.520·33-s + 0.0945·35-s + 0.0660·37-s + 0.478·39-s − 1.17·41-s − 0.0744·43-s − 0.494·45-s + 0.283·47-s + 0.142·49-s + 1.77·51-s + 1.01·53-s + 0.0754·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4564896832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4564896832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 + 0.559T + 5T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 - 2.00T + 19T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 + 2.70T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 - 0.402T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 + 0.488T + 43T^{2} \) |
| 47 | \( 1 - 1.94T + 47T^{2} \) |
| 53 | \( 1 - 7.42T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 5.74T + 61T^{2} \) |
| 67 | \( 1 - 8.52T + 67T^{2} \) |
| 71 | \( 1 + 7.30T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 + 2.61T + 79T^{2} \) |
| 83 | \( 1 + 0.759T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.49793178800900796248637732785, −6.98311524160266541909396350276, −6.49678937847567019944505404911, −5.58732337905042288774743932899, −5.26550366798328745371916545061, −4.45145714479255063123591177087, −3.75682454896127193790851055378, −2.63320329502732504302736872953, −1.48022394041595948593135674714, −0.37855907474488165649068681878,
0.37855907474488165649068681878, 1.48022394041595948593135674714, 2.63320329502732504302736872953, 3.75682454896127193790851055378, 4.45145714479255063123591177087, 5.26550366798328745371916545061, 5.58732337905042288774743932899, 6.49678937847567019944505404911, 6.98311524160266541909396350276, 7.49793178800900796248637732785
