L(s) = 1 | − 3·5-s + 7-s − 7·13-s + 2·17-s + 19-s + 6·23-s + 4·25-s − 7·29-s − 10·31-s − 3·35-s − 7·37-s − 6·41-s + 8·43-s − 13·47-s + 49-s − 10·53-s − 9·59-s + 6·61-s + 21·65-s + 67-s + 6·71-s − 13·73-s − 8·79-s − 18·83-s − 6·85-s + 2·89-s − 7·91-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.377·7-s − 1.94·13-s + 0.485·17-s + 0.229·19-s + 1.25·23-s + 4/5·25-s − 1.29·29-s − 1.79·31-s − 0.507·35-s − 1.15·37-s − 0.937·41-s + 1.21·43-s − 1.89·47-s + 1/7·49-s − 1.37·53-s − 1.17·59-s + 0.768·61-s + 2.60·65-s + 0.122·67-s + 0.712·71-s − 1.52·73-s − 0.900·79-s − 1.97·83-s − 0.650·85-s + 0.211·89-s − 0.733·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.91340044105715, −14.43332116531180, −14.00359183630351, −13.03320482201835, −12.68112056154042, −12.33451244138647, −11.68761075782772, −11.32552062017393, −10.92198351398729, −10.23965752469431, −9.593104993163013, −9.219774777946471, −8.549021629794155, −7.918213859128748, −7.498851854352673, −7.170013338701291, −6.705164022841895, −5.464083087981123, −5.342085671653135, −4.606672932598485, −4.135508176738912, −3.297624753680307, −3.032510322860639, −2.014238531547959, −1.372518518792123, 0, 0,
1.372518518792123, 2.014238531547959, 3.032510322860639, 3.297624753680307, 4.135508176738912, 4.606672932598485, 5.342085671653135, 5.464083087981123, 6.705164022841895, 7.170013338701291, 7.498851854352673, 7.918213859128748, 8.549021629794155, 9.219774777946471, 9.593104993163013, 10.23965752469431, 10.92198351398729, 11.32552062017393, 11.68761075782772, 12.33451244138647, 12.68112056154042, 13.03320482201835, 14.00359183630351, 14.43332116531180, 14.91340044105715