| L(s) = 1 | − 2-s + 4-s + 1.06·5-s + 2.41·7-s − 8-s − 1.06·10-s + 2.87·11-s + 6.31·13-s − 2.41·14-s + 16-s + 5.80·17-s + 2.51·19-s + 1.06·20-s − 2.87·22-s + 4.87·23-s − 3.86·25-s − 6.31·26-s + 2.41·28-s + 5.18·29-s − 7.91·31-s − 32-s − 5.80·34-s + 2.57·35-s − 4.64·37-s − 2.51·38-s − 1.06·40-s + 10.6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.477·5-s + 0.912·7-s − 0.353·8-s − 0.337·10-s + 0.867·11-s + 1.75·13-s − 0.644·14-s + 0.250·16-s + 1.40·17-s + 0.577·19-s + 0.238·20-s − 0.613·22-s + 1.01·23-s − 0.772·25-s − 1.23·26-s + 0.456·28-s + 0.962·29-s − 1.42·31-s − 0.176·32-s − 0.995·34-s + 0.435·35-s − 0.763·37-s − 0.408·38-s − 0.168·40-s + 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.629177526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.629177526\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 1.06T + 5T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 - 6.31T + 13T^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 - 2.51T + 19T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 - 5.18T + 29T^{2} \) |
| 31 | \( 1 + 7.91T + 31T^{2} \) |
| 37 | \( 1 + 4.64T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 + 2.68T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
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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
−7.78454567856038064743862341856, −7.38829353704707657793698966131, −6.43230303838675870174565818241, −5.79704511381062995291620105428, −5.31373658088635829192491670705, −4.11719281944565495443849491682, −3.51710140571995602602576659086, −2.49353153984483218173765013266, −1.24340086217206695307026680668, −1.20847776283533113375550546205,
1.20847776283533113375550546205, 1.24340086217206695307026680668, 2.49353153984483218173765013266, 3.51710140571995602602576659086, 4.11719281944565495443849491682, 5.31373658088635829192491670705, 5.79704511381062995291620105428, 6.43230303838675870174565818241, 7.38829353704707657793698966131, 7.78454567856038064743862341856
