L(s) = 1 | − 2-s + 4-s − 8-s − 4·13-s + 16-s + 6·17-s − 4·19-s − 6·23-s − 5·25-s + 4·26-s + 6·29-s − 8·31-s − 32-s − 6·34-s − 10·37-s + 4·38-s − 6·41-s − 8·43-s + 6·46-s − 6·47-s + 5·50-s − 4·52-s − 6·58-s + 8·61-s + 8·62-s + 64-s − 4·67-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 1.25·23-s − 25-s + 0.784·26-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 1.64·37-s + 0.648·38-s − 0.937·41-s − 1.21·43-s + 0.884·46-s − 0.875·47-s + 0.707·50-s − 0.554·52-s − 0.787·58-s + 1.02·61-s + 1.01·62-s + 1/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.35232916706859, −13.69471386308613, −13.21651735936986, −12.45527940996748, −12.12404935343590, −11.88553182151534, −11.24568087627675, −10.47455624011694, −10.24194764763729, −9.784951126606731, −9.423728168672662, −8.547711086224653, −8.309902762996197, −7.797058215925956, −7.214604453259703, −6.791296070053912, −6.201057444577284, −5.487727711801215, −5.209094606600183, −4.392692740649945, −3.702829000635981, −3.232603061797970, −2.444328916939911, −1.834275457766010, −1.360277424921043, 0, 0,
1.360277424921043, 1.834275457766010, 2.444328916939911, 3.232603061797970, 3.702829000635981, 4.392692740649945, 5.209094606600183, 5.487727711801215, 6.201057444577284, 6.791296070053912, 7.214604453259703, 7.797058215925956, 8.309902762996197, 8.547711086224653, 9.423728168672662, 9.784951126606731, 10.24194764763729, 10.47455624011694, 11.24568087627675, 11.88553182151534, 12.12404935343590, 12.45527940996748, 13.21651735936986, 13.69471386308613, 14.35232916706859