L(s) = 1 | + (0.951 − 0.309i)2-s + (0.951 + 1.30i)3-s + (0.809 − 0.587i)4-s + (1.30 + 0.951i)6-s − 0.618i·7-s + (0.587 − 0.809i)8-s + (−0.500 + 1.53i)9-s + (1.53 + 0.499i)12-s + (−0.190 − 0.587i)14-s + (0.309 − 0.951i)16-s + 1.61i·18-s + (0.809 − 0.587i)21-s + (−0.587 + 0.190i)23-s + 1.61·24-s + (−0.951 + 0.309i)27-s + (−0.363 − 0.5i)28-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.951 + 1.30i)3-s + (0.809 − 0.587i)4-s + (1.30 + 0.951i)6-s − 0.618i·7-s + (0.587 − 0.809i)8-s + (−0.500 + 1.53i)9-s + (1.53 + 0.499i)12-s + (−0.190 − 0.587i)14-s + (0.309 − 0.951i)16-s + 1.61i·18-s + (0.809 − 0.587i)21-s + (−0.587 + 0.190i)23-s + 1.61·24-s + (−0.951 + 0.309i)27-s + (−0.363 − 0.5i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.898716914\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.898716914\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + 0.618iT - T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61iT - T^{2} \) |
| 47 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (1.17 - 1.61i)T + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.951i)T^{2} \) |
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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
−9.294203374629806473327494652144, −8.543960628949269464631633030538, −7.58308873787033789162465821992, −6.84923060842189288630458980759, −5.68543196872876400824109904909, −4.97666201745309866740223588922, −4.12813530428516596067613470016, −3.64672147612338442872376753403, −2.86916960390537221918179305969, −1.75891071309678734283969457490,
1.68780907719614261726436948462, 2.40053026532362934309339420900, 3.20257953901204852852976298966, 4.13441781221625964230617081179, 5.28769019559688693368630369661, 6.15090698484985619484777970437, 6.65667482433622299051378460598, 7.71193514535986602020273284743, 7.85865846640175284293798801494, 8.792401217365312320452771138282