| L(s) = 1 | + 5.37·3-s + 26.4·7-s + 1.86·9-s − 49.8·11-s + 49.0·13-s − 17.2·17-s − 19·19-s + 142.·21-s − 166.·23-s − 135.·27-s − 109.·29-s − 273.·31-s − 267.·33-s − 167.·37-s + 263.·39-s + 15.1·41-s − 413.·43-s + 161.·47-s + 358.·49-s − 92.5·51-s + 490.·53-s − 102.·57-s − 335.·59-s + 725.·61-s + 49.3·63-s − 497.·67-s − 896.·69-s + ⋯ |
| L(s) = 1 | + 1.03·3-s + 1.43·7-s + 0.0689·9-s − 1.36·11-s + 1.04·13-s − 0.245·17-s − 0.229·19-s + 1.47·21-s − 1.51·23-s − 0.962·27-s − 0.699·29-s − 1.58·31-s − 1.41·33-s − 0.742·37-s + 1.08·39-s + 0.0577·41-s − 1.46·43-s + 0.501·47-s + 1.04·49-s − 0.254·51-s + 1.27·53-s − 0.237·57-s − 0.741·59-s + 1.52·61-s + 0.0986·63-s − 0.907·67-s − 1.56·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 5.37T + 27T^{2} \) |
| 7 | \( 1 - 26.4T + 343T^{2} \) |
| 11 | \( 1 + 49.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 49.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 15.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 161.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 335.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 497.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 798.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 311.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 665.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 372.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 960.T + 9.12e5T^{2} \) |
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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
−8.456182413356147791074543701814, −7.88429342549613025545692486921, −7.26053426305269897391457388509, −5.85513693394061842813657387541, −5.27235565489910892017466236436, −4.18916808861639182118533886021, −3.39837192155630217109660304442, −2.23908139978217311750195556569, −1.69278694563732583830108948070, 0,
1.69278694563732583830108948070, 2.23908139978217311750195556569, 3.39837192155630217109660304442, 4.18916808861639182118533886021, 5.27235565489910892017466236436, 5.85513693394061842813657387541, 7.26053426305269897391457388509, 7.88429342549613025545692486921, 8.456182413356147791074543701814
