L(s) = 1 | − 1.45·2-s + 0.115·4-s − 0.945·5-s − 4.13·7-s + 2.74·8-s + 1.37·10-s − 5.73·11-s + 2.28·13-s + 6.00·14-s − 4.21·16-s − 0.810·17-s + 19-s − 0.109·20-s + 8.34·22-s − 5.93·23-s − 4.10·25-s − 3.33·26-s − 0.477·28-s − 0.945·29-s + 5.55·31-s + 0.653·32-s + 1.17·34-s + 3.90·35-s + 11.4·37-s − 1.45·38-s − 2.59·40-s + 4.86·41-s + ⋯ |
L(s) = 1 | − 1.02·2-s + 0.0578·4-s − 0.422·5-s − 1.56·7-s + 0.969·8-s + 0.434·10-s − 1.72·11-s + 0.635·13-s + 1.60·14-s − 1.05·16-s − 0.196·17-s + 0.229·19-s − 0.0244·20-s + 1.77·22-s − 1.23·23-s − 0.821·25-s − 0.653·26-s − 0.0902·28-s − 0.175·29-s + 0.996·31-s + 0.115·32-s + 0.202·34-s + 0.659·35-s + 1.87·37-s − 0.235·38-s − 0.409·40-s + 0.759·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 5 | \( 1 + 0.945T + 5T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 + 0.810T + 17T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 + 0.945T + 29T^{2} \) |
| 31 | \( 1 - 5.55T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 53 | \( 1 - 8.62T + 53T^{2} \) |
| 59 | \( 1 - 3.15T + 59T^{2} \) |
| 61 | \( 1 - 6.40T + 61T^{2} \) |
| 67 | \( 1 - 5.94T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 0.0686T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 7.86T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−7.64021457252626474614858445294, −7.09379528282880775862621432738, −6.07973074512195563351231554458, −5.67510360340988618688799461951, −4.48959525847366609681635006824, −3.91579176346493455058926471765, −2.93979955163160064566083745188, −2.23292591794087052928288680278, −0.77336096131854196266535136785, 0,
0.77336096131854196266535136785, 2.23292591794087052928288680278, 2.93979955163160064566083745188, 3.91579176346493455058926471765, 4.48959525847366609681635006824, 5.67510360340988618688799461951, 6.07973074512195563351231554458, 7.09379528282880775862621432738, 7.64021457252626474614858445294