| L(s) = 1 | − 5-s + 4·7-s − 6·13-s + 6·17-s + 4·19-s + 25-s + 2·29-s + 8·31-s − 4·35-s − 10·37-s + 6·41-s + 43-s − 8·47-s + 9·49-s − 6·53-s − 10·61-s + 6·65-s + 4·67-s − 8·71-s − 6·73-s + 16·83-s − 6·85-s − 6·89-s − 24·91-s − 4·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.676·35-s − 1.64·37-s + 0.937·41-s + 0.152·43-s − 1.16·47-s + 9/7·49-s − 0.824·53-s − 1.28·61-s + 0.744·65-s + 0.488·67-s − 0.949·71-s − 0.702·73-s + 1.75·83-s − 0.650·85-s − 0.635·89-s − 2.51·91-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.98274101162400, −13.40444703566513, −12.55190861064759, −12.22777119797404, −11.74665441707969, −11.69339527391861, −10.77887154149647, −10.50466355197341, −9.815575611006249, −9.544607393996751, −8.807916057345525, −8.122407450037948, −7.944734513632575, −7.422562902280014, −7.080777026832181, −6.274709465352690, −5.566845124689949, −5.037022315955983, −4.793830713463864, −4.259960158860800, −3.373414432270525, −2.945131301749858, −2.237863438085264, −1.480726187129530, −0.9900802452588899, 0,
0.9900802452588899, 1.480726187129530, 2.237863438085264, 2.945131301749858, 3.373414432270525, 4.259960158860800, 4.793830713463864, 5.037022315955983, 5.566845124689949, 6.274709465352690, 7.080777026832181, 7.422562902280014, 7.944734513632575, 8.122407450037948, 8.807916057345525, 9.544607393996751, 9.815575611006249, 10.50466355197341, 10.77887154149647, 11.69339527391861, 11.74665441707969, 12.22777119797404, 12.55190861064759, 13.40444703566513, 13.98274101162400
