| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 12-s − 2·13-s + 14-s + 16-s − 6·17-s − 18-s − 2·19-s − 21-s + 23-s − 24-s − 5·25-s + 2·26-s + 27-s − 28-s − 31-s − 32-s + 6·34-s + 36-s − 4·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.218·21-s + 0.208·23-s − 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.179·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.657·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−13.99969740760010, −13.56608220043192, −13.11117429053334, −12.73323052416592, −12.04096235416897, −11.59832491189644, −11.16498842580830, −10.49106957671372, −10.12759511933988, −9.575495348186614, −9.264181429663331, −8.603220225397512, −8.277363191314317, −7.811932683261328, −7.072915237766622, −6.683470979391236, −6.390445844098919, −5.536009891458023, −4.860289912777990, −4.434184495654539, −3.495404359946009, −3.292751321467969, −2.356010590358272, −1.987839561141296, −1.378983421847602, 0, 0,
1.378983421847602, 1.987839561141296, 2.356010590358272, 3.292751321467969, 3.495404359946009, 4.434184495654539, 4.860289912777990, 5.536009891458023, 6.390445844098919, 6.683470979391236, 7.072915237766622, 7.811932683261328, 8.277363191314317, 8.603220225397512, 9.264181429663331, 9.575495348186614, 10.12759511933988, 10.49106957671372, 11.16498842580830, 11.59832491189644, 12.04096235416897, 12.73323052416592, 13.11117429053334, 13.56608220043192, 13.99969740760010
