L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.382 − 0.923i)3-s + 1.00i·4-s + (−0.923 − 0.382i)5-s + (−0.382 + 0.923i)6-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.382 + 0.923i)10-s + (0.923 − 0.382i)12-s + (−0.541 − 0.541i)13-s + 1.00·14-s + i·15-s − 1.00·16-s + 18-s + 1.84i·19-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.382 − 0.923i)3-s + 1.00i·4-s + (−0.923 − 0.382i)5-s + (−0.382 + 0.923i)6-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.382 + 0.923i)10-s + (0.923 − 0.382i)12-s + (−0.541 − 0.541i)13-s + 1.00·14-s + i·15-s − 1.00·16-s + 18-s + 1.84i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1750864545\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1750864545\) |
\(L(1)\) |
|
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 | \( 1 + (0.382 + 0.923i)T \) |
| 5 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - 1.84iT - T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 1.84T + T^{2} \) |
| 61 | \( 1 + 0.765T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 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
−10.56866998043723203331269948086, −9.745259742073124331476742790291, −8.774779484700329283491748978111, −7.913852965275893498573504110748, −7.56182946060276040619216196599, −6.34602570806304002307911038068, −5.35279101507200909719597279142, −3.88832938884724760510556981396, −2.88604215080063581166308535490, −1.57622486052582564016848014234,
0.23268649900585883291428216113, 2.86112427347970306147167813418, 4.22760948806766480507821910609, 4.79037348430907972409233403756, 6.20720151604004699083869255318, 6.83324538811742872264712317638, 7.62882154370374393745749445059, 8.716768499566425109567626270583, 9.418670159911763129464192980516, 10.22567940190814304387754338166