L(s) = 1 | − 2·5-s − 4·11-s + 2·13-s + 17-s + 4·19-s − 25-s + 10·29-s + 8·31-s − 2·37-s + 10·41-s − 12·43-s − 6·53-s + 8·55-s − 12·59-s + 10·61-s − 4·65-s + 12·67-s − 10·73-s + 8·79-s − 4·83-s − 2·85-s − 6·89-s − 8·95-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 1.20·11-s + 0.554·13-s + 0.242·17-s + 0.917·19-s − 1/5·25-s + 1.85·29-s + 1.43·31-s − 0.328·37-s + 1.56·41-s − 1.82·43-s − 0.824·53-s + 1.07·55-s − 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.46·67-s − 1.17·73-s + 0.900·79-s − 0.439·83-s − 0.216·85-s − 0.635·89-s − 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.76290032923239, −13.39425423285423, −12.82776450997335, −12.16207678104578, −12.03497241079254, −11.35919785619048, −10.98921370001590, −10.41771079424869, −9.912900554866983, −9.592173927959116, −8.672912967698311, −8.323711523365018, −7.905284461946741, −7.558296642249657, −6.834275966393962, −6.365222840160314, −5.764163476452228, −5.057634106546271, −4.758856657404687, −4.076095393461321, −3.460795437002302, −2.903723088815798, −2.484835405192891, −1.426486749010713, −0.8150248070500647, 0,
0.8150248070500647, 1.426486749010713, 2.484835405192891, 2.903723088815798, 3.460795437002302, 4.076095393461321, 4.758856657404687, 5.057634106546271, 5.764163476452228, 6.365222840160314, 6.834275966393962, 7.558296642249657, 7.905284461946741, 8.323711523365018, 8.672912967698311, 9.592173927959116, 9.912900554866983, 10.41771079424869, 10.98921370001590, 11.35919785619048, 12.03497241079254, 12.16207678104578, 12.82776450997335, 13.39425423285423, 13.76290032923239