L(s) = 1 | + (0.222 − 0.974i)2-s + (−0.777 + 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)5-s + (0.777 + 0.974i)6-s + (0.900 + 0.433i)7-s + (−0.623 + 0.781i)8-s + (−0.123 − 0.541i)9-s + (−0.623 − 0.781i)10-s + (1.12 − 0.541i)12-s + (0.623 − 0.781i)14-s + (0.277 + 1.21i)15-s + (0.623 + 0.781i)16-s − 0.554·18-s + (−0.900 + 0.433i)20-s + (−1.12 + 0.541i)21-s + ⋯ |
L(s) = 1 | + (0.222 − 0.974i)2-s + (−0.777 + 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)5-s + (0.777 + 0.974i)6-s + (0.900 + 0.433i)7-s + (−0.623 + 0.781i)8-s + (−0.123 − 0.541i)9-s + (−0.623 − 0.781i)10-s + (1.12 − 0.541i)12-s + (0.623 − 0.781i)14-s + (0.277 + 1.21i)15-s + (0.623 + 0.781i)16-s − 0.554·18-s + (−0.900 + 0.433i)20-s + (−1.12 + 0.541i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9788126551\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9788126551\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.222 + 0.974i)T \) |
| 5 | \( 1 + (-0.623 + 0.781i)T \) |
| 7 | \( 1 + (-0.900 - 0.433i)T \) |
good | 3 | \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 29 | \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 41 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 61 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.34530991924404823930953305060, −9.452155667858362759447104990376, −8.872306565612576551839565997798, −7.968528680854625806827841626455, −6.18666433641034223173661907239, −5.33408840265871537983279562309, −4.86252003664822354211519184532, −4.15529939304290375261098484025, −2.64405627866272526515290270100, −1.34097608348028343016065956745,
1.30851887438826301727517610117, 2.97369842387203172125456815135, 4.46578934803074059405648017685, 5.36912787626619798347866556839, 6.18522970786729600504910007904, 6.91241138038862862238823909852, 7.36184117479901622369655356246, 8.333835363881519608793803056839, 9.283468218780219889094901487094, 10.43424981927449387413532568064