L(s) = 1 | − 1.72·2-s − 29.0·4-s − 73.1·5-s + 105.·8-s + 125.·10-s − 193.·11-s − 151.·13-s + 747.·16-s + 1.76e3·17-s − 1.12e3·19-s + 2.12e3·20-s + 332.·22-s − 2.41e3·23-s + 2.22e3·25-s + 261.·26-s + 7.72e3·29-s + 8.17e3·31-s − 4.65e3·32-s − 3.03e3·34-s + 5.93e3·37-s + 1.94e3·38-s − 7.68e3·40-s + 4.97e3·41-s − 5.65e3·43-s + 5.60e3·44-s + 4.15e3·46-s + 4.72e3·47-s + ⋯ |
L(s) = 1 | − 0.304·2-s − 0.907·4-s − 1.30·5-s + 0.580·8-s + 0.398·10-s − 0.481·11-s − 0.248·13-s + 0.730·16-s + 1.47·17-s − 0.716·19-s + 1.18·20-s + 0.146·22-s − 0.951·23-s + 0.711·25-s + 0.0757·26-s + 1.70·29-s + 1.52·31-s − 0.803·32-s − 0.449·34-s + 0.712·37-s + 0.218·38-s − 0.759·40-s + 0.462·41-s − 0.466·43-s + 0.436·44-s + 0.289·46-s + 0.311·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.72T + 32T^{2} \) |
| 5 | \( 1 + 73.1T + 3.12e3T^{2} \) |
| 11 | \( 1 + 193.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 151.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.76e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.97e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.65e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.72e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.20e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.85e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.90e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.72e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.06e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.34e5T + 8.58e9T^{2} \) |
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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
−10.01726790493519846061698238014, −8.750612711784064299887338665389, −8.027919406858157817466543334811, −7.53911855263057160498065521026, −6.04884527078206248051322163018, −4.76595998783021677919650717204, −4.09165414951758407511774919519, −2.94014478315800471154889557590, −1.00216270888322709782583949528, 0,
1.00216270888322709782583949528, 2.94014478315800471154889557590, 4.09165414951758407511774919519, 4.76595998783021677919650717204, 6.04884527078206248051322163018, 7.53911855263057160498065521026, 8.027919406858157817466543334811, 8.750612711784064299887338665389, 10.01726790493519846061698238014