| L(s) = 1 | − 2-s + 0.745·3-s + 4-s + 2.52·5-s − 0.745·6-s + 4.14·7-s − 8-s − 2.44·9-s − 2.52·10-s − 11-s + 0.745·12-s + 0.317·13-s − 4.14·14-s + 1.88·15-s + 16-s − 0.311·17-s + 2.44·18-s + 4.80·19-s + 2.52·20-s + 3.09·21-s + 22-s + 4.58·23-s − 0.745·24-s + 1.39·25-s − 0.317·26-s − 4.06·27-s + 4.14·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.430·3-s + 0.5·4-s + 1.13·5-s − 0.304·6-s + 1.56·7-s − 0.353·8-s − 0.814·9-s − 0.799·10-s − 0.301·11-s + 0.215·12-s + 0.0881·13-s − 1.10·14-s + 0.487·15-s + 0.250·16-s − 0.0755·17-s + 0.575·18-s + 1.10·19-s + 0.565·20-s + 0.675·21-s + 0.213·22-s + 0.955·23-s − 0.152·24-s + 0.279·25-s − 0.0623·26-s − 0.781·27-s + 0.784·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.412543512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.412543512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.745T + 3T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 13 | \( 1 - 0.317T + 13T^{2} \) |
| 17 | \( 1 + 0.311T + 17T^{2} \) |
| 19 | \( 1 - 4.80T + 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 + 5.00T + 37T^{2} \) |
| 41 | \( 1 - 8.13T + 41T^{2} \) |
| 43 | \( 1 - 0.409T + 43T^{2} \) |
| 47 | \( 1 - 2.81T + 47T^{2} \) |
| 53 | \( 1 + 5.22T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 3.43T + 61T^{2} \) |
| 67 | \( 1 - 7.85T + 67T^{2} \) |
| 71 | \( 1 + 8.35T + 71T^{2} \) |
| 73 | \( 1 - 7.35T + 73T^{2} \) |
| 79 | \( 1 - 8.91T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 2.11T + 89T^{2} \) |
| 97 | \( 1 - 6.62T + 97T^{2} \) |
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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
−8.382194424316923839980080025361, −7.81775551556787155698021144205, −7.20327725429598460268296045462, −6.10166529315483171613326485661, −5.44583938351655758891334003699, −4.93013380853304808701839673386, −3.58680599766717841985954848373, −2.53307036026416386793122552660, −1.96009974262439671074515545923, −1.00248144582410270611269589350,
1.00248144582410270611269589350, 1.96009974262439671074515545923, 2.53307036026416386793122552660, 3.58680599766717841985954848373, 4.93013380853304808701839673386, 5.44583938351655758891334003699, 6.10166529315483171613326485661, 7.20327725429598460268296045462, 7.81775551556787155698021144205, 8.382194424316923839980080025361
