| L(s) = 1 | + 1.73·2-s + 0.999·4-s + 0.456·5-s − 3.79·7-s − 1.73·8-s + 0.791·10-s − 2.18·11-s − 1.20·13-s − 6.56·14-s − 5·16-s − 4.37·17-s + 3.58·19-s + 0.456·20-s − 3.79·22-s − 4.79·25-s − 2.09·26-s − 3.79·28-s − 3.10·29-s + 3.58·31-s − 5.19·32-s − 7.58·34-s − 1.73·35-s − 2.79·37-s + 6.20·38-s − 0.791·40-s + 9.66·41-s − 8.37·43-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s + 0.204·5-s − 1.43·7-s − 0.612·8-s + 0.250·10-s − 0.659·11-s − 0.335·13-s − 1.75·14-s − 1.25·16-s − 1.06·17-s + 0.821·19-s + 0.102·20-s − 0.808·22-s − 0.958·25-s − 0.410·26-s − 0.716·28-s − 0.576·29-s + 0.643·31-s − 0.918·32-s − 1.30·34-s − 0.292·35-s − 0.458·37-s + 1.00·38-s − 0.125·40-s + 1.51·41-s − 1.27·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 - 0.456T + 5T^{2} \) |
| 7 | \( 1 + 3.79T + 7T^{2} \) |
| 11 | \( 1 + 2.18T + 11T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 3.10T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 + 2.79T + 37T^{2} \) |
| 41 | \( 1 - 9.66T + 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 + 0.456T + 47T^{2} \) |
| 53 | \( 1 + 3.10T + 53T^{2} \) |
| 59 | \( 1 - 8.75T + 59T^{2} \) |
| 61 | \( 1 - 1.79T + 61T^{2} \) |
| 67 | \( 1 + 1.20T + 67T^{2} \) |
| 71 | \( 1 + 0.361T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 5.58T + 79T^{2} \) |
| 83 | \( 1 - 0.723T + 83T^{2} \) |
| 89 | \( 1 + 8.29T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.547024555522333256545061657439, −8.711341375383994535082030190084, −7.45469852692897971459370615712, −6.57706693474649267734109910281, −5.90427637480831382634663310109, −5.10330258454646037471373975942, −4.09832273940973271868053900073, −3.21440899850369878407693683711, −2.39649192370244019337362006160, 0,
2.39649192370244019337362006160, 3.21440899850369878407693683711, 4.09832273940973271868053900073, 5.10330258454646037471373975942, 5.90427637480831382634663310109, 6.57706693474649267734109910281, 7.45469852692897971459370615712, 8.711341375383994535082030190084, 9.547024555522333256545061657439
