| L(s) = 1 | − 2-s + 4-s + 1.06·5-s + 4.42·7-s − 8-s − 1.06·10-s + 1.64·11-s + 5.47·13-s − 4.42·14-s + 16-s − 3.92·17-s + 6.60·19-s + 1.06·20-s − 1.64·22-s − 3.86·25-s − 5.47·26-s + 4.42·28-s + 6.39·29-s − 3.01·31-s − 32-s + 3.92·34-s + 4.72·35-s − 2.21·37-s − 6.60·38-s − 1.06·40-s − 4.09·41-s − 2.91·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.477·5-s + 1.67·7-s − 0.353·8-s − 0.337·10-s + 0.497·11-s + 1.51·13-s − 1.18·14-s + 0.250·16-s − 0.952·17-s + 1.51·19-s + 0.238·20-s − 0.351·22-s − 0.772·25-s − 1.07·26-s + 0.836·28-s + 1.18·29-s − 0.541·31-s − 0.176·32-s + 0.673·34-s + 0.798·35-s − 0.363·37-s − 1.07·38-s − 0.168·40-s − 0.639·41-s − 0.444·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.571084383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.571084383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 1.06T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 3.92T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 29 | \( 1 - 6.39T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 37 | \( 1 + 2.21T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 + 2.91T + 43T^{2} \) |
| 47 | \( 1 - 9.62T + 47T^{2} \) |
| 53 | \( 1 - 2.23T + 53T^{2} \) |
| 59 | \( 1 + 9.75T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 5.26T + 71T^{2} \) |
| 73 | \( 1 - 7.18T + 73T^{2} \) |
| 79 | \( 1 - 2.69T + 79T^{2} \) |
| 83 | \( 1 + 5.53T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.87354078073438295205647229044, −7.10607291662347305976193555836, −6.41490213077684630799800378011, −5.67705296384463916451504852840, −5.06312283245388071485893124233, −4.19669524881364006331853329443, −3.40537007711745447059116752980, −2.26234495127503023048640808945, −1.55878269568674134423000938303, −0.958423239692940930912202482832,
0.958423239692940930912202482832, 1.55878269568674134423000938303, 2.26234495127503023048640808945, 3.40537007711745447059116752980, 4.19669524881364006331853329443, 5.06312283245388071485893124233, 5.67705296384463916451504852840, 6.41490213077684630799800378011, 7.10607291662347305976193555836, 7.87354078073438295205647229044
