| L(s) = 1 | − 2.41·2-s − 1.12·3-s + 3.84·4-s + 1.14·5-s + 2.71·6-s + 1.36·7-s − 4.46·8-s − 1.74·9-s − 2.75·10-s + 0.0870·11-s − 4.31·12-s + 6.43·13-s − 3.29·14-s − 1.27·15-s + 3.09·16-s + 2.91·17-s + 4.21·18-s + 3.65·19-s + 4.38·20-s − 1.52·21-s − 0.210·22-s − 0.109·23-s + 5.00·24-s − 3.69·25-s − 15.5·26-s + 5.31·27-s + 5.24·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.647·3-s + 1.92·4-s + 0.509·5-s + 1.10·6-s + 0.515·7-s − 1.57·8-s − 0.581·9-s − 0.871·10-s + 0.0262·11-s − 1.24·12-s + 1.78·13-s − 0.881·14-s − 0.330·15-s + 0.774·16-s + 0.706·17-s + 0.993·18-s + 0.838·19-s + 0.980·20-s − 0.333·21-s − 0.0448·22-s − 0.0227·23-s + 1.02·24-s − 0.739·25-s − 3.05·26-s + 1.02·27-s + 0.990·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8719256980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8719256980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4027 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 1.12T + 3T^{2} \) |
| 5 | \( 1 - 1.14T + 5T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 - 0.0870T + 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 + 0.109T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + 8.84T + 37T^{2} \) |
| 41 | \( 1 - 8.26T + 41T^{2} \) |
| 43 | \( 1 + 4.98T + 43T^{2} \) |
| 47 | \( 1 - 5.18T + 47T^{2} \) |
| 53 | \( 1 - 4.14T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 3.54T + 61T^{2} \) |
| 67 | \( 1 - 1.69T + 67T^{2} \) |
| 71 | \( 1 - 7.48T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 - 2.88T + 97T^{2} \) |
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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
−8.570140106053370727338738000964, −7.983750255199362287157580481719, −7.12813829448234237730088102031, −6.27223285878839783207946556401, −5.84918705825917672640058139896, −4.96599853969506961398275916958, −3.60867271242610687350671453300, −2.55973663446218090351897127357, −1.43275060568729910729084609492, −0.803255964942521368104286790114,
0.803255964942521368104286790114, 1.43275060568729910729084609492, 2.55973663446218090351897127357, 3.60867271242610687350671453300, 4.96599853969506961398275916958, 5.84918705825917672640058139896, 6.27223285878839783207946556401, 7.12813829448234237730088102031, 7.983750255199362287157580481719, 8.570140106053370727338738000964
