L(s) = 1 | + 5-s − 4·7-s + 4·11-s − 6·17-s + 4·23-s + 25-s + 6·29-s + 8·31-s − 4·35-s + 2·37-s + 10·41-s − 4·43-s + 8·47-s + 9·49-s + 2·53-s + 4·55-s + 4·59-s + 14·61-s + 12·67-s − 8·71-s + 10·73-s − 16·77-s − 4·83-s − 6·85-s + 10·89-s + 2·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.20·11-s − 1.45·17-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.676·35-s + 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.539·55-s + 0.520·59-s + 1.79·61-s + 1.46·67-s − 0.949·71-s + 1.17·73-s − 1.82·77-s − 0.439·83-s − 0.650·85-s + 1.05·89-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.561427384\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.561427384\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.26430931310369, −13.67217404536830, −13.29113428471685, −12.89728579683111, −12.27641424240929, −11.90287127301077, −11.16626212769778, −10.79685817886628, −9.990601394026714, −9.735647923294735, −9.207876591407692, −8.728418107226493, −8.306771960267357, −7.254485671893503, −6.825792425887078, −6.404947805414369, −6.130766824657105, −5.298621971535807, −4.556570430207797, −4.033701346043248, −3.457447526173431, −2.582482621499014, −2.388320068550518, −1.141300428400326, −0.6187591656158080,
0.6187591656158080, 1.141300428400326, 2.388320068550518, 2.582482621499014, 3.457447526173431, 4.033701346043248, 4.556570430207797, 5.298621971535807, 6.130766824657105, 6.404947805414369, 6.825792425887078, 7.254485671893503, 8.306771960267357, 8.728418107226493, 9.207876591407692, 9.735647923294735, 9.990601394026714, 10.79685817886628, 11.16626212769778, 11.90287127301077, 12.27641424240929, 12.89728579683111, 13.29113428471685, 13.67217404536830, 14.26430931310369