| L(s) = 1 | − 1.04·2-s − 9·3-s − 30.8·4-s + 80.7·5-s + 9.44·6-s + 79.1·7-s + 66.0·8-s + 81·9-s − 84.7·10-s − 97.2·11-s + 278.·12-s − 49.3·13-s − 83.0·14-s − 727.·15-s + 919.·16-s − 1.98e3·17-s − 85.0·18-s − 1.54e3·19-s − 2.49e3·20-s − 712.·21-s + 102.·22-s + 1.04e3·23-s − 594.·24-s + 3.40e3·25-s + 51.8·26-s − 729·27-s − 2.44e3·28-s + ⋯ |
| L(s) = 1 | − 0.185·2-s − 0.577·3-s − 0.965·4-s + 1.44·5-s + 0.107·6-s + 0.610·7-s + 0.364·8-s + 0.333·9-s − 0.268·10-s − 0.242·11-s + 0.557·12-s − 0.0810·13-s − 0.113·14-s − 0.834·15-s + 0.897·16-s − 1.66·17-s − 0.0618·18-s − 0.981·19-s − 1.39·20-s − 0.352·21-s + 0.0449·22-s + 0.410·23-s − 0.210·24-s + 1.08·25-s + 0.0150·26-s − 0.192·27-s − 0.589·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 + 1.04T + 32T^{2} \) |
| 5 | \( 1 - 80.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 79.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 97.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 49.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.98e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.54e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.49e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.80e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.99e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.91e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.16e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.70e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.41e5T + 8.58e9T^{2} \) |
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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
−9.781886985047036197076769568674, −8.986148963869009387876036653152, −8.206655117673899333446285213265, −6.79741383516674580206401435089, −5.92361510828890955836545541225, −4.99089910035020851357529775709, −4.31452056404225196335071363811, −2.40967946566281887742276840310, −1.34037607166899018708922665117, 0,
1.34037607166899018708922665117, 2.40967946566281887742276840310, 4.31452056404225196335071363811, 4.99089910035020851357529775709, 5.92361510828890955836545541225, 6.79741383516674580206401435089, 8.206655117673899333446285213265, 8.986148963869009387876036653152, 9.781886985047036197076769568674
