L(s) = 1 | + 3.53·2-s + 4.46·4-s − 2.07·5-s − 12.4·8-s − 7.33·10-s − 49.1·11-s + 44.8·13-s − 79.7·16-s + 26.5·17-s − 77.7·19-s − 9.28·20-s − 173.·22-s − 55.7·23-s − 120.·25-s + 158.·26-s − 121.·29-s − 305.·31-s − 181.·32-s + 93.6·34-s + 77.1·37-s − 274.·38-s + 25.9·40-s + 248.·41-s − 147.·43-s − 219.·44-s − 196.·46-s + 269.·47-s + ⋯ |
L(s) = 1 | + 1.24·2-s + 0.558·4-s − 0.185·5-s − 0.551·8-s − 0.231·10-s − 1.34·11-s + 0.956·13-s − 1.24·16-s + 0.378·17-s − 0.938·19-s − 0.103·20-s − 1.68·22-s − 0.505·23-s − 0.965·25-s + 1.19·26-s − 0.777·29-s − 1.77·31-s − 1.00·32-s + 0.472·34-s + 0.342·37-s − 1.17·38-s + 0.102·40-s + 0.947·41-s − 0.521·43-s − 0.753·44-s − 0.630·46-s + 0.837·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.53T + 8T^{2} \) |
| 5 | \( 1 + 2.07T + 125T^{2} \) |
| 11 | \( 1 + 49.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 55.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 305.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 77.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 141.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 424.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 179.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 237.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 495.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 24.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.66e3T + 9.12e5T^{2} \) |
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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
−10.56519199392465973000591506581, −9.343439763784883234499669333651, −8.320117958379788358074059890287, −7.36150491684451516819093226457, −6.01660965866939631524963455906, −5.48649233971671423605556176053, −4.28097477324558481757172951234, −3.46430297096013593242428796931, −2.19028283652367587780518388921, 0,
2.19028283652367587780518388921, 3.46430297096013593242428796931, 4.28097477324558481757172951234, 5.48649233971671423605556176053, 6.01660965866939631524963455906, 7.36150491684451516819093226457, 8.320117958379788358074059890287, 9.343439763784883234499669333651, 10.56519199392465973000591506581