L(s) = 1 | + 2·13-s − 8·19-s − 5·25-s − 4·31-s + 10·37-s + 8·43-s + 14·61-s − 16·67-s + 10·73-s + 4·79-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.554·13-s − 1.83·19-s − 25-s − 0.718·31-s + 1.64·37-s + 1.21·43-s + 1.79·61-s − 1.95·67-s + 1.17·73-s + 0.450·79-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.46821528320345, −14.91173069378384, −14.50329902722202, −13.90411500315802, −13.20794468560532, −12.92674105639394, −12.37602627202987, −11.63357499264642, −11.19733968919178, −10.63601645062315, −10.19192470965491, −9.366668611309472, −9.052111428031050, −8.250002425519016, −7.931848058418903, −7.171098751085239, −6.504936316197059, −5.987628313104290, −5.503271996771148, −4.511139990908336, −4.124621445637085, −3.478589533669437, −2.493298667434967, −2.010390926931963, −1.015397726855132, 0,
1.015397726855132, 2.010390926931963, 2.493298667434967, 3.478589533669437, 4.124621445637085, 4.511139990908336, 5.503271996771148, 5.987628313104290, 6.504936316197059, 7.171098751085239, 7.931848058418903, 8.250002425519016, 9.052111428031050, 9.366668611309472, 10.19192470965491, 10.63601645062315, 11.19733968919178, 11.63357499264642, 12.37602627202987, 12.92674105639394, 13.20794468560532, 13.90411500315802, 14.50329902722202, 14.91173069378384, 15.46821528320345