L(s) = 1 | + 16·2-s + 256·4-s − 1.18e3·5-s + 1.17e3·7-s + 4.09e3·8-s − 1.89e4·10-s − 1.46e4·11-s + 8.24e4·13-s + 1.87e4·14-s + 6.55e4·16-s − 4.40e5·17-s + 8.04e5·19-s − 3.03e5·20-s − 2.34e5·22-s + 2.28e5·23-s − 5.47e5·25-s + 1.31e6·26-s + 3.00e5·28-s − 2.28e6·29-s + 2.56e6·31-s + 1.04e6·32-s − 7.05e6·34-s − 1.39e6·35-s − 4.68e6·37-s + 1.28e7·38-s − 4.85e6·40-s − 3.30e6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.848·5-s + 0.184·7-s + 0.353·8-s − 0.599·10-s − 0.301·11-s + 0.800·13-s + 0.130·14-s + 0.250·16-s − 1.28·17-s + 1.41·19-s − 0.424·20-s − 0.213·22-s + 0.170·23-s − 0.280·25-s + 0.566·26-s + 0.0924·28-s − 0.599·29-s + 0.498·31-s + 0.176·32-s − 0.905·34-s − 0.156·35-s − 0.411·37-s + 1.00·38-s − 0.299·40-s − 0.182·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 16T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 1.46e4T \) |
good | 5 | \( 1 + 1.18e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.17e3T + 4.03e7T^{2} \) |
| 13 | \( 1 - 8.24e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.40e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.28e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.56e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.68e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.30e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.81e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.85e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.97e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.31e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.12e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.27e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.55e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.96e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.29e9T + 7.60e17T^{2} \) |
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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
−10.71858799463182567963262821577, −9.292219758533295041001609635090, −8.119676511713394713550796783069, −7.26301568925839135192570903278, −6.11289864116375646139058154253, −4.92625497950434354924992429821, −3.94725522397425187339296481114, −2.94190004612741043061326678379, −1.47701974900509870972813318162, 0,
1.47701974900509870972813318162, 2.94190004612741043061326678379, 3.94725522397425187339296481114, 4.92625497950434354924992429821, 6.11289864116375646139058154253, 7.26301568925839135192570903278, 8.119676511713394713550796783069, 9.292219758533295041001609635090, 10.71858799463182567963262821577