L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s − 2·13-s + 16-s + 2·17-s − 4·19-s − 20-s − 22-s + 4·23-s + 25-s + 2·26-s + 6·29-s − 32-s − 2·34-s + 2·37-s + 4·38-s + 40-s + 6·41-s + 12·43-s + 44-s − 4·46-s + 8·47-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.176·32-s − 0.342·34-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + 1.82·43-s + 0.150·44-s − 0.589·46-s + 1.16·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.778339421\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.778339421\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.66732858067037, −14.25223492842685, −13.50723258554694, −12.85922791428212, −12.38034125040386, −12.02603861465864, −11.33169057115188, −10.94932574323759, −10.33401248664860, −9.956948754840582, −9.173743649879669, −8.891349036660239, −8.245277410491918, −7.748969713554426, −7.141330252023590, −6.778452821929981, −6.029928246559551, −5.486011285697012, −4.695128308965675, −4.118739741156643, −3.496695524594288, −2.544089859294263, −2.302562146519531, −1.046672362957583, −0.6455843197834562,
0.6455843197834562, 1.046672362957583, 2.302562146519531, 2.544089859294263, 3.496695524594288, 4.118739741156643, 4.695128308965675, 5.486011285697012, 6.029928246559551, 6.778452821929981, 7.141330252023590, 7.748969713554426, 8.245277410491918, 8.891349036660239, 9.173743649879669, 9.956948754840582, 10.33401248664860, 10.94932574323759, 11.33169057115188, 12.02603861465864, 12.38034125040386, 12.85922791428212, 13.50723258554694, 14.25223492842685, 14.66732858067037