| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 3·11-s − 13-s + 2·14-s + 16-s − 8·17-s − 20-s − 3·22-s − 4·23-s − 4·25-s + 26-s − 2·28-s + 29-s − 3·31-s − 32-s + 8·34-s + 2·35-s + 8·37-s + 40-s − 2·41-s − 11·43-s + 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.904·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.94·17-s − 0.223·20-s − 0.639·22-s − 0.834·23-s − 4/5·25-s + 0.196·26-s − 0.377·28-s + 0.185·29-s − 0.538·31-s − 0.176·32-s + 1.37·34-s + 0.338·35-s + 1.31·37-s + 0.158·40-s − 0.312·41-s − 1.67·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.26866147802001152913025958268, −9.482258294877269881590757644436, −8.763894772751692239273117298832, −7.81012507517906630443430989651, −6.74819217569927040729981669784, −6.19818150900186047644659279635, −4.54621693456857955686454044680, −3.46305229707620473692544260201, −2.01064091456439163113006432735, 0,
2.01064091456439163113006432735, 3.46305229707620473692544260201, 4.54621693456857955686454044680, 6.19818150900186047644659279635, 6.74819217569927040729981669784, 7.81012507517906630443430989651, 8.763894772751692239273117298832, 9.482258294877269881590757644436, 10.26866147802001152913025958268
