| L(s) = 1 | − 2-s + 1.33·3-s + 4-s − 3.04·5-s − 1.33·6-s − 8-s − 1.21·9-s + 3.04·10-s − 1.14·11-s + 1.33·12-s + 2.19·13-s − 4.07·15-s + 16-s + 2.75·17-s + 1.21·18-s − 5.40·19-s − 3.04·20-s + 1.14·22-s + 2.31·23-s − 1.33·24-s + 4.27·25-s − 2.19·26-s − 5.63·27-s + 3.73·29-s + 4.07·30-s − 3.26·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.771·3-s + 0.5·4-s − 1.36·5-s − 0.545·6-s − 0.353·8-s − 0.404·9-s + 0.962·10-s − 0.345·11-s + 0.385·12-s + 0.608·13-s − 1.05·15-s + 0.250·16-s + 0.668·17-s + 0.285·18-s − 1.23·19-s − 0.680·20-s + 0.244·22-s + 0.483·23-s − 0.272·24-s + 0.854·25-s − 0.430·26-s − 1.08·27-s + 0.693·29-s + 0.743·30-s − 0.587·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 1.33T + 3T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 17 | \( 1 - 2.75T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 - 2.31T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 + 3.26T + 31T^{2} \) |
| 37 | \( 1 - 5.59T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 - 4.42T + 43T^{2} \) |
| 47 | \( 1 - 8.76T + 47T^{2} \) |
| 53 | \( 1 + 1.28T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 + 0.292T + 61T^{2} \) |
| 67 | \( 1 - 0.230T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 1.92T + 73T^{2} \) |
| 83 | \( 1 - 7.41T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 9.87T + 97T^{2} \) |
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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
−7.70727645523106626724829018600, −7.22318717768918829060239100607, −6.24259850332030130373066903468, −5.57066086887451840098670445348, −4.36274683668338577802854180229, −3.84217357284611533395417597400, −2.98717827334555263038225477050, −2.40023630808505815814532311822, −1.08924725019230489289766047421, 0,
1.08924725019230489289766047421, 2.40023630808505815814532311822, 2.98717827334555263038225477050, 3.84217357284611533395417597400, 4.36274683668338577802854180229, 5.57066086887451840098670445348, 6.24259850332030130373066903468, 7.22318717768918829060239100607, 7.70727645523106626724829018600
