| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 4·11-s + 4·13-s − 14-s + 16-s + 5·17-s − 6·19-s − 20-s − 4·22-s − 23-s + 25-s − 4·26-s + 28-s − 31-s − 32-s − 5·34-s − 35-s − 2·37-s + 6·38-s + 40-s + 2·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s − 1.37·19-s − 0.223·20-s − 0.852·22-s − 0.208·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s − 0.179·31-s − 0.176·32-s − 0.857·34-s − 0.169·35-s − 0.328·37-s + 0.973·38-s + 0.158·40-s + 0.312·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.33329821955897, −13.99100827845935, −13.23416339049040, −12.72624217303446, −12.10880292225139, −11.80280950489748, −11.19552247470826, −10.90439679420733, −10.20761533576759, −9.858542965104723, −9.073573350152058, −8.703105681522200, −8.299786334176091, −7.799215831855518, −7.162537933295296, −6.604378213453548, −6.159935803299040, −5.608392906214236, −4.816680991141445, −4.093295714130738, −3.681661768696436, −3.103602473406164, −2.149338432912841, −1.479720227190783, −1.008655134360410, 0,
1.008655134360410, 1.479720227190783, 2.149338432912841, 3.103602473406164, 3.681661768696436, 4.093295714130738, 4.816680991141445, 5.608392906214236, 6.159935803299040, 6.604378213453548, 7.162537933295296, 7.799215831855518, 8.299786334176091, 8.703105681522200, 9.073573350152058, 9.858542965104723, 10.20761533576759, 10.90439679420733, 11.19552247470826, 11.80280950489748, 12.10880292225139, 12.72624217303446, 13.23416339049040, 13.99100827845935, 14.33329821955897
