L(s) = 1 | + (0.707 + 0.707i)2-s + i·3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + (−0.707 + 0.707i)8-s − 9-s − 1.00·10-s − 1.00·12-s + (−0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1 + i)17-s + (−0.707 − 0.707i)18-s + (−0.707 − 0.707i)20-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + i·3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + (−0.707 + 0.707i)8-s − 9-s − 1.00·10-s − 1.00·12-s + (−0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1 + i)17-s + (−0.707 − 0.707i)18-s + (−0.707 − 0.707i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.175483695\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.175483695\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 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
−10.23006855854908685978643745859, −9.181460371540066783284521825432, −8.035419189884655035287114731924, −7.914896106717085390732882620065, −6.67614029781394825009949343312, −5.93931269872367222149634940592, −5.01866581593170072494926261589, −4.26530475305542719394515418278, −3.36033630425602045094744731035, −2.78567125768940623919766862562,
0.74045425211659851367426791725, 1.98583808909537944933566435786, 3.03073712906181149412029151633, 4.06919068177439757629953234196, 5.06808002884782947531880488265, 5.67056732946157086730293372168, 6.89651324810535105516244453191, 7.39349888205057143726986260400, 8.466103488921207574889386950426, 9.197607796521076055857795209308