| L(s) = 1 | + 2-s − 4-s + 5-s − 7-s − 3·8-s + 10-s − 14-s − 16-s − 2·17-s + 8·19-s − 20-s − 8·23-s + 25-s + 28-s + 2·29-s − 4·31-s + 5·32-s − 2·34-s − 35-s + 2·37-s + 8·38-s − 3·40-s − 6·41-s + 4·43-s − 8·46-s + 8·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s − 1.06·8-s + 0.316·10-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 1.83·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s + 0.188·28-s + 0.371·29-s − 0.718·31-s + 0.883·32-s − 0.342·34-s − 0.169·35-s + 0.328·37-s + 1.29·38-s − 0.474·40-s − 0.937·41-s + 0.609·43-s − 1.17·46-s + 1.16·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.40117105699923, −14.05940739051098, −13.84113522032060, −13.19183804594112, −12.82034668119024, −12.21718089095066, −11.78589671060630, −11.34685141198004, −10.43772207935189, −10.02279075293136, −9.564153631618671, −9.015607965655655, −8.636301605642988, −7.741995851998353, −7.419819786733593, −6.526927633565031, −6.028837894164765, −5.643204069459522, −4.993499461861907, −4.486416262447264, −3.756345630198824, −3.308657827377643, −2.633078248201369, −1.862221201432503, −0.9124962622457804, 0,
0.9124962622457804, 1.862221201432503, 2.633078248201369, 3.308657827377643, 3.756345630198824, 4.486416262447264, 4.993499461861907, 5.643204069459522, 6.028837894164765, 6.526927633565031, 7.419819786733593, 7.741995851998353, 8.636301605642988, 9.015607965655655, 9.564153631618671, 10.02279075293136, 10.43772207935189, 11.34685141198004, 11.78589671060630, 12.21718089095066, 12.82034668119024, 13.19183804594112, 13.84113522032060, 14.05940739051098, 14.40117105699923
