L(s) = 1 | + (0.382 − 0.923i)2-s + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)4-s + (0.382 + 0.923i)5-s + (−0.382 − 0.923i)6-s + (−0.923 + 0.382i)8-s − 1.00i·9-s + 10-s + 1.84·11-s − 12-s + (−0.707 − 0.707i)13-s + (0.923 + 0.382i)15-s + i·16-s + (−0.923 − 0.382i)18-s + (0.382 − 0.923i)20-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)2-s + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)4-s + (0.382 + 0.923i)5-s + (−0.382 − 0.923i)6-s + (−0.923 + 0.382i)8-s − 1.00i·9-s + 10-s + 1.84·11-s − 12-s + (−0.707 − 0.707i)13-s + (0.923 + 0.382i)15-s + i·16-s + (−0.923 − 0.382i)18-s + (0.382 − 0.923i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.706838420\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.706838420\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.382 + 0.923i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.382 - 0.923i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - 1.84T + T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + 0.765T + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + 0.765iT - T^{2} \) |
| 61 | \( 1 + 1.41iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.84iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 89 | \( 1 - 1.84iT - T^{2} \) |
| 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
−9.672926871366788754052993046703, −8.783788774053758882625286404041, −7.890970895323825489636608064479, −6.71500639445408121061876998796, −6.39844294626630074887524130756, −5.20449026315102783991792841271, −3.88045409738657295570573515846, −3.26046804360944876249085346011, −2.31629495240296061589942592590, −1.35983823549704205282968685797,
1.80554722541465229422251812685, 3.33201940402259172931230067895, 4.25516670060208125221873554307, 4.71087504475759015018107314330, 5.69582846550419139298342504425, 6.63026848944941631373951053946, 7.44993073406774760372663424894, 8.496465999426475443062466039735, 8.978762594906878339744490270988, 9.458993324297811627968807479621