L(s) = 1 | + 5-s − 3·9-s − 2·13-s − 2·17-s + 8·19-s + 8·23-s + 25-s − 6·29-s + 2·37-s + 6·41-s − 8·43-s − 3·45-s + 8·47-s + 2·53-s + 8·59-s − 2·61-s − 2·65-s − 8·67-s − 10·73-s − 16·79-s + 9·81-s + 16·83-s − 2·85-s − 10·89-s + 8·95-s + 14·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 9-s − 0.554·13-s − 0.485·17-s + 1.83·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.328·37-s + 0.937·41-s − 1.21·43-s − 0.447·45-s + 1.16·47-s + 0.274·53-s + 1.04·59-s − 0.256·61-s − 0.248·65-s − 0.977·67-s − 1.17·73-s − 1.80·79-s + 81-s + 1.75·83-s − 0.216·85-s − 1.05·89-s + 0.820·95-s + 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.028394086\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.028394086\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.07454872562768, −15.37495624423736, −14.71832157293250, −14.48384833808787, −13.68870355372905, −13.31457452661716, −12.76364024210033, −11.89878793395456, −11.55794761211032, −11.00281846315610, −10.33983877748685, −9.651365153864231, −9.066720848023466, −8.807600632538271, −7.783559092213748, −7.342820090844085, −6.708985428270152, −5.847206416502694, −5.378938997680072, −4.886674922766744, −3.907759322705510, −2.999125151405839, −2.658061460331654, −1.587585674329236, −0.6226994877106028,
0.6226994877106028, 1.587585674329236, 2.658061460331654, 2.999125151405839, 3.907759322705510, 4.886674922766744, 5.378938997680072, 5.847206416502694, 6.708985428270152, 7.342820090844085, 7.783559092213748, 8.807600632538271, 9.066720848023466, 9.651365153864231, 10.33983877748685, 11.00281846315610, 11.55794761211032, 11.89878793395456, 12.76364024210033, 13.31457452661716, 13.68870355372905, 14.48384833808787, 14.71832157293250, 15.37495624423736, 16.07454872562768