| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s − 2·13-s − 16-s − 2·17-s + 4·19-s − 20-s + 25-s + 2·26-s + 6·29-s − 5·32-s + 2·34-s + 6·37-s − 4·38-s + 3·40-s + 6·41-s + 4·43-s − 50-s + 2·52-s + 2·53-s − 6·58-s + 4·59-s + 6·61-s + 7·64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s − 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.648·38-s + 0.474·40-s + 0.937·41-s + 0.609·43-s − 0.141·50-s + 0.277·52-s + 0.274·53-s − 0.787·58-s + 0.520·59-s + 0.768·61-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.97122294341897, −12.64856230352827, −12.14777099881266, −11.50159717611218, −11.12163786865382, −10.60376806807169, −10.09656693454187, −9.675807306585493, −9.476347958887394, −8.820055967948164, −8.534339032642169, −7.815256609304592, −7.641970680545963, −6.933888201055277, −6.562661755786423, −5.845989764423115, −5.376562940435201, −4.858365885009050, −4.436507656920521, −3.858646683477617, −3.229504467805876, −2.388671150077877, −2.206318075884978, −1.054113901653388, −0.9543699116005574, 0,
0.9543699116005574, 1.054113901653388, 2.206318075884978, 2.388671150077877, 3.229504467805876, 3.858646683477617, 4.436507656920521, 4.858365885009050, 5.376562940435201, 5.845989764423115, 6.562661755786423, 6.933888201055277, 7.641970680545963, 7.815256609304592, 8.534339032642169, 8.820055967948164, 9.476347958887394, 9.675807306585493, 10.09656693454187, 10.60376806807169, 11.12163786865382, 11.50159717611218, 12.14777099881266, 12.64856230352827, 12.97122294341897
