| L(s) = 1 | − 3-s + 2·5-s + 9-s + 6·13-s − 2·15-s + 6·17-s − 8·19-s − 25-s − 27-s + 6·29-s + 6·37-s − 6·39-s − 10·41-s + 8·43-s + 2·45-s − 6·51-s + 6·53-s + 8·57-s − 4·59-s − 2·61-s + 12·65-s − 12·67-s − 8·71-s + 2·73-s + 75-s + 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.66·13-s − 0.516·15-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.986·37-s − 0.960·39-s − 1.56·41-s + 1.21·43-s + 0.298·45-s − 0.840·51-s + 0.824·53-s + 1.05·57-s − 0.520·59-s − 0.256·61-s + 1.48·65-s − 1.46·67-s − 0.949·71-s + 0.234·73-s + 0.115·75-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.804655275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.804655275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.42102364903307, −13.00296243656812, −12.36103029761060, −12.07219656575637, −11.43185231310236, −10.92877016277719, −10.39201173035844, −10.26730318128957, −9.630951622903885, −9.004689294631506, −8.574781163255403, −8.106795476168184, −7.531768305594634, −6.786214142082172, −6.306704276200757, −5.955135402896157, −5.660667149852144, −4.932327144741893, −4.283050238703367, −3.865422429209140, −3.121693126546324, −2.510158840821647, −1.672938190769604, −1.315382182768022, −0.5368937813500083,
0.5368937813500083, 1.315382182768022, 1.672938190769604, 2.510158840821647, 3.121693126546324, 3.865422429209140, 4.283050238703367, 4.932327144741893, 5.660667149852144, 5.955135402896157, 6.306704276200757, 6.786214142082172, 7.531768305594634, 8.106795476168184, 8.574781163255403, 9.004689294631506, 9.630951622903885, 10.26730318128957, 10.39201173035844, 10.92877016277719, 11.43185231310236, 12.07219656575637, 12.36103029761060, 13.00296243656812, 13.42102364903307
