| L(s) = 1 | − 5-s + 4·7-s + 4·11-s + 2·13-s − 17-s − 4·19-s − 4·23-s + 25-s + 2·29-s − 4·31-s − 4·35-s + 6·37-s − 2·41-s − 12·43-s + 8·47-s + 9·49-s − 2·53-s − 4·55-s − 12·59-s − 2·61-s − 2·65-s + 4·67-s − 4·71-s − 14·73-s + 16·77-s + 12·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 1.20·11-s + 0.554·13-s − 0.242·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.676·35-s + 0.986·37-s − 0.312·41-s − 1.82·43-s + 1.16·47-s + 9/7·49-s − 0.274·53-s − 0.539·55-s − 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.488·67-s − 0.474·71-s − 1.63·73-s + 1.82·77-s + 1.35·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.79558098155377, −14.40826320431038, −13.82242336103356, −13.41661358677385, −12.64988283361120, −12.05942138234862, −11.73604282081450, −11.22178827847548, −10.81436722770020, −10.28866912758262, −9.505869862208261, −8.865549622449343, −8.559022073015689, −7.951734532291308, −7.573938889784973, −6.756751442774723, −6.318901393929916, −5.687780398067160, −4.910741739323524, −4.380266822751130, −4.000523145464574, −3.313805501071233, −2.320515648144411, −1.650854889795661, −1.167321214397388, 0,
1.167321214397388, 1.650854889795661, 2.320515648144411, 3.313805501071233, 4.000523145464574, 4.380266822751130, 4.910741739323524, 5.687780398067160, 6.318901393929916, 6.756751442774723, 7.573938889784973, 7.951734532291308, 8.559022073015689, 8.865549622449343, 9.505869862208261, 10.28866912758262, 10.81436722770020, 11.22178827847548, 11.73604282081450, 12.05942138234862, 12.64988283361120, 13.41661358677385, 13.82242336103356, 14.40826320431038, 14.79558098155377
