| L(s) = 1 | − 5-s + 7-s − 4·11-s − 5·13-s + 8·17-s + 5·19-s + 8·23-s + 25-s + 3·29-s + 31-s − 35-s + 8·37-s + 11·41-s + 43-s + 10·47-s − 6·49-s + 2·53-s + 4·55-s + 3·61-s + 5·65-s − 11·67-s − 6·71-s − 11·73-s − 4·77-s + 7·79-s − 8·85-s + 16·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s − 1.38·13-s + 1.94·17-s + 1.14·19-s + 1.66·23-s + 1/5·25-s + 0.557·29-s + 0.179·31-s − 0.169·35-s + 1.31·37-s + 1.71·41-s + 0.152·43-s + 1.45·47-s − 6/7·49-s + 0.274·53-s + 0.539·55-s + 0.384·61-s + 0.620·65-s − 1.34·67-s − 0.712·71-s − 1.28·73-s − 0.455·77-s + 0.787·79-s − 0.867·85-s + 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.319762538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.319762538\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.05794580672635, −14.50595760354455, −14.29974984180021, −13.44785563614332, −12.93364718284548, −12.37905437126239, −11.98507052048273, −11.43859789930672, −10.76743831239264, −10.30848501831650, −9.704613232211912, −9.284051394320486, −8.478902460388285, −7.740866094999591, −7.488766684236265, −7.236850851325157, −6.042501168538947, −5.526544102645372, −4.923300003263480, −4.588114826486185, −3.557837818888938, −2.790334797897311, −2.609872453639040, −1.234565669969843, −0.6569361068696943,
0.6569361068696943, 1.234565669969843, 2.609872453639040, 2.790334797897311, 3.557837818888938, 4.588114826486185, 4.923300003263480, 5.526544102645372, 6.042501168538947, 7.236850851325157, 7.488766684236265, 7.740866094999591, 8.478902460388285, 9.284051394320486, 9.704613232211912, 10.30848501831650, 10.76743831239264, 11.43859789930672, 11.98507052048273, 12.37905437126239, 12.93364718284548, 13.44785563614332, 14.29974984180021, 14.50595760354455, 15.05794580672635
