L(s) = 1 | + 2·5-s − 7-s − 2·13-s − 2·17-s + 4·19-s + 8·23-s − 25-s − 2·29-s − 4·31-s − 2·35-s + 6·37-s + 6·41-s + 4·43-s + 12·47-s + 49-s + 10·53-s − 8·59-s − 10·61-s − 4·65-s + 4·67-s + 8·71-s + 2·73-s + 8·79-s + 12·83-s − 4·85-s − 10·89-s + 2·91-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.338·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 1.37·53-s − 1.04·59-s − 1.28·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 0.234·73-s + 0.900·79-s + 1.31·83-s − 0.433·85-s − 1.05·89-s + 0.209·91-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
\(L(1)\) |
\(\approx\) |
\(2.944460667\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.944460667\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.08201453835262, −13.87277329489516, −13.28519830287800, −12.82174241452396, −12.41549272023897, −11.78937296700435, −11.12421164580172, −10.76455058132129, −10.20322755601525, −9.483609212003478, −9.259190073865850, −8.942330782220712, −7.973036965190888, −7.382435815776008, −7.099869675445421, −6.294917084560235, −5.856829997103700, −5.309531887783272, −4.785456659225920, −4.036920094741975, −3.375964675130138, −2.571576588580641, −2.286704074099409, −1.286519370658432, −0.6158024708816638,
0.6158024708816638, 1.286519370658432, 2.286704074099409, 2.571576588580641, 3.375964675130138, 4.036920094741975, 4.785456659225920, 5.309531887783272, 5.856829997103700, 6.294917084560235, 7.099869675445421, 7.382435815776008, 7.973036965190888, 8.942330782220712, 9.259190073865850, 9.483609212003478, 10.20322755601525, 10.76455058132129, 11.12421164580172, 11.78937296700435, 12.41549272023897, 12.82174241452396, 13.28519830287800, 13.87277329489516, 14.08201453835262