| L(s) = 1 | − 5-s − 4·7-s − 13-s + 4·17-s − 4·19-s − 8·23-s + 25-s − 29-s + 6·31-s + 4·35-s + 37-s − 10·41-s + 5·43-s − 7·47-s + 9·49-s − 9·53-s − 3·59-s + 4·61-s + 65-s − 8·67-s + 6·71-s − 4·73-s − 6·79-s + 9·83-s − 4·85-s + 7·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.277·13-s + 0.970·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.185·29-s + 1.07·31-s + 0.676·35-s + 0.164·37-s − 1.56·41-s + 0.762·43-s − 1.02·47-s + 9/7·49-s − 1.23·53-s − 0.390·59-s + 0.512·61-s + 0.124·65-s − 0.977·67-s + 0.712·71-s − 0.468·73-s − 0.675·79-s + 0.987·83-s − 0.433·85-s + 0.741·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−13.76084059572541, −13.51756050552345, −12.80507726835573, −12.52284394271503, −12.03931956741664, −11.67269145120273, −10.98557909385408, −10.28446165376387, −10.00906976647960, −9.707991215823692, −8.978959874305153, −8.502835151561075, −7.854496822909824, −7.580917066054085, −6.754575136596456, −6.359980421524752, −6.032760950972067, −5.321841537433813, −4.588357701040597, −4.107703203584233, −3.358003955158784, −3.185307018190218, −2.333716466490372, −1.671481973982290, −0.6233341049372148, 0,
0.6233341049372148, 1.671481973982290, 2.333716466490372, 3.185307018190218, 3.358003955158784, 4.107703203584233, 4.588357701040597, 5.321841537433813, 6.032760950972067, 6.359980421524752, 6.754575136596456, 7.580917066054085, 7.854496822909824, 8.502835151561075, 8.978959874305153, 9.707991215823692, 10.00906976647960, 10.28446165376387, 10.98557909385408, 11.67269145120273, 12.03931956741664, 12.52284394271503, 12.80507726835573, 13.51756050552345, 13.76084059572541
