| L(s) = 1 | − 3·9-s − 6·11-s + 4·13-s − 2·17-s − 8·19-s − 8·23-s − 6·29-s − 6·31-s + 2·37-s − 6·41-s − 4·43-s + 4·47-s − 7·49-s + 6·53-s − 6·59-s − 61-s − 12·67-s − 2·71-s + 4·73-s − 10·79-s + 9·81-s + 4·83-s − 10·89-s + 8·97-s + 18·99-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 9-s − 1.80·11-s + 1.10·13-s − 0.485·17-s − 1.83·19-s − 1.66·23-s − 1.11·29-s − 1.07·31-s + 0.328·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 49-s + 0.824·53-s − 0.781·59-s − 0.128·61-s − 1.46·67-s − 0.237·71-s + 0.468·73-s − 1.12·79-s + 81-s + 0.439·83-s − 1.05·89-s + 0.812·97-s + 1.80·99-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−15.97804089117475, −15.30213335244705, −14.99215993171855, −14.36158919239759, −13.60149811187434, −13.32308547126079, −12.83433485133433, −12.21260839387456, −11.47689279172441, −10.96132907920924, −10.60572015826593, −10.10602324283951, −9.229815560372542, −8.629278560479562, −8.223592632269519, −7.798561208222678, −6.975897270862897, −6.133497569023332, −5.850806105373780, −5.232273684004442, −4.407150222365896, −3.773138748236400, −3.027493074405108, −2.255345508929801, −1.754731961069739, 0, 0,
1.754731961069739, 2.255345508929801, 3.027493074405108, 3.773138748236400, 4.407150222365896, 5.232273684004442, 5.850806105373780, 6.133497569023332, 6.975897270862897, 7.798561208222678, 8.223592632269519, 8.629278560479562, 9.229815560372542, 10.10602324283951, 10.60572015826593, 10.96132907920924, 11.47689279172441, 12.21260839387456, 12.83433485133433, 13.32308547126079, 13.60149811187434, 14.36158919239759, 14.99215993171855, 15.30213335244705, 15.97804089117475
