L(s) = 1 | + (1 − i)2-s − i·4-s + i·5-s + i·9-s + (1 + i)10-s + 11-s + (−1 − i)13-s + 16-s + (1 + i)18-s + i·19-s + 20-s + (1 − i)22-s + (−1 − i)23-s − 25-s − 2·26-s + ⋯ |
L(s) = 1 | + (1 − i)2-s − i·4-s + i·5-s + i·9-s + (1 + i)10-s + 11-s + (−1 − i)13-s + 16-s + (1 + i)18-s + i·19-s + 20-s + (1 − i)22-s + (−1 − i)23-s − 25-s − 2·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.736474603\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.736474603\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - iT \) |
good | 2 | \( 1 + (-1 + i)T - iT^{2} \) |
| 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + 2iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1 - i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.18370987162743641083590069238, −9.854345473161023048654013660687, −8.119966507835763453819939323957, −7.68440188662252246586365449730, −6.40259518971876131588421356605, −5.60710096546892564815887618524, −4.53478897405041601349273461536, −3.75173983637425831397079934493, −2.69728799695975153235870854607, −1.98953361268774909305785756183,
1.50623806758096587503904121300, 3.46975110841508378527812456417, 4.30616758053099226860088634974, 4.98632450209736878733032099082, 5.89629164892603253984312620578, 6.78224605934323013604861424701, 7.27972448965396821559649864185, 8.512757720099854588943270119186, 9.292075785195061160221044845461, 9.808446244840076116337404848435