L(s) = 1 | + (0.809 + 0.587i)3-s + (−0.587 − 0.809i)4-s + (0.951 + 0.309i)5-s + (0.309 + 0.951i)9-s − i·12-s + (0.587 + 0.809i)15-s + (−0.309 + 0.951i)16-s + (−0.309 − 0.951i)20-s + (1 − i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)36-s + (−0.221 + 1.39i)37-s + i·45-s + (−0.221 − 1.39i)47-s + (−0.809 + 0.587i)48-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)3-s + (−0.587 − 0.809i)4-s + (0.951 + 0.309i)5-s + (0.309 + 0.951i)9-s − i·12-s + (0.587 + 0.809i)15-s + (−0.309 + 0.951i)16-s + (−0.309 − 0.951i)20-s + (1 − i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (0.587 − 0.809i)36-s + (−0.221 + 1.39i)37-s + i·45-s + (−0.221 − 1.39i)47-s + (−0.809 + 0.587i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.555592044\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.555592044\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.951 - 0.309i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 7 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 13 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.221 - 1.39i)T + (-0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (1.26 + 0.642i)T + (0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1.26 + 0.642i)T + (0.587 + 0.809i)T^{2} \) |
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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
−9.659931170311677122256644141167, −8.780503433999691187032248966150, −8.333270392995470950256078618650, −7.00264167406011327446125980420, −6.28932581246320633314964909409, −5.19192692271523757042086050970, −4.78720653206249052695182942515, −3.59368072634075987018227040342, −2.57349987353603872920628121981, −1.53651721688494831589490319082,
1.34510550730812747594953797276, 2.55154363168585647105837136708, 3.34438948103381383593618295508, 4.37134983524369575372985704929, 5.34332689805520977921846453142, 6.33014284832622970555972625890, 7.26527262219690005890254434658, 7.86081497491826048482662559566, 8.758246782531909228936729470602, 9.242165880296988833364905813983