L(s) = 1 | + 2-s − 1.75·3-s + 4-s + 1.28·5-s − 1.75·6-s − 0.876·7-s + 8-s + 0.0641·9-s + 1.28·10-s − 1.28·11-s − 1.75·12-s + 0.225·13-s − 0.876·14-s − 2.25·15-s + 16-s + 5.98·17-s + 0.0641·18-s − 7.24·19-s + 1.28·20-s + 1.53·21-s − 1.28·22-s + 2.70·23-s − 1.75·24-s − 3.33·25-s + 0.225·26-s + 5.13·27-s − 0.876·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.01·3-s + 0.5·4-s + 0.576·5-s − 0.714·6-s − 0.331·7-s + 0.353·8-s + 0.0213·9-s + 0.407·10-s − 0.386·11-s − 0.505·12-s + 0.0624·13-s − 0.234·14-s − 0.582·15-s + 0.250·16-s + 1.45·17-s + 0.0151·18-s − 1.66·19-s + 0.288·20-s + 0.334·21-s − 0.273·22-s + 0.565·23-s − 0.357·24-s − 0.667·25-s + 0.0441·26-s + 0.989·27-s − 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 0.876T + 7T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 - 0.225T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 + 6.42T + 29T^{2} \) |
| 31 | \( 1 - 1.49T + 31T^{2} \) |
| 37 | \( 1 + 2.10T + 37T^{2} \) |
| 41 | \( 1 - 4.13T + 41T^{2} \) |
| 43 | \( 1 + 4.91T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 - 2.06T + 53T^{2} \) |
| 59 | \( 1 + 8.60T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 - 2.50T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 6.40T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 8.26T + 97T^{2} \) |
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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
−7.953982221562760447344037161663, −7.05610041162609535189008128457, −6.32174121335212453274276253378, −5.76238266924674237889338446610, −5.31521170914075155739014691715, −4.43628892688633030519354731547, −3.48002905837383944613712059609, −2.56406008251426114337064052667, −1.48065817134941534416421858055, 0,
1.48065817134941534416421858055, 2.56406008251426114337064052667, 3.48002905837383944613712059609, 4.43628892688633030519354731547, 5.31521170914075155739014691715, 5.76238266924674237889338446610, 6.32174121335212453274276253378, 7.05610041162609535189008128457, 7.953982221562760447344037161663