| L(s) = 1 | − 0.614·3-s + 5-s + 2.24·7-s − 2.62·9-s + 5.19·13-s − 0.614·15-s − 3.22·17-s + 7.19·19-s − 1.38·21-s − 5.11·23-s + 25-s + 3.45·27-s + 9.61·29-s − 0.491·31-s + 2.24·35-s + 0.350·37-s − 3.19·39-s − 4.42·41-s + 10.9·43-s − 2.62·45-s + 1.03·47-s − 1.95·49-s + 1.98·51-s + 3.75·53-s − 4.42·57-s − 12.7·59-s − 6.38·61-s + ⋯ |
| L(s) = 1 | − 0.355·3-s + 0.447·5-s + 0.848·7-s − 0.873·9-s + 1.44·13-s − 0.158·15-s − 0.783·17-s + 1.65·19-s − 0.301·21-s − 1.06·23-s + 0.200·25-s + 0.665·27-s + 1.78·29-s − 0.0882·31-s + 0.379·35-s + 0.0576·37-s − 0.511·39-s − 0.690·41-s + 1.67·43-s − 0.390·45-s + 0.150·47-s − 0.279·49-s + 0.278·51-s + 0.515·53-s − 0.586·57-s − 1.65·59-s − 0.818·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.355809059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.355809059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.614T + 3T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 + 5.11T + 23T^{2} \) |
| 29 | \( 1 - 9.61T + 29T^{2} \) |
| 31 | \( 1 + 0.491T + 31T^{2} \) |
| 37 | \( 1 - 0.350T + 37T^{2} \) |
| 41 | \( 1 + 4.42T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 1.03T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 - 8.44T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 2.13T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 9.90T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.86954311299612431794326407575, −6.85784332567369613649223781990, −6.17552078642226912621702421708, −5.71541974557285538754161035881, −5.01345446703866898173053257845, −4.31905349931525681949643675658, −3.37238641848443861950431248987, −2.60221139490202071013830942343, −1.60536645389069388044071939879, −0.788881557675966881591240100463,
0.788881557675966881591240100463, 1.60536645389069388044071939879, 2.60221139490202071013830942343, 3.37238641848443861950431248987, 4.31905349931525681949643675658, 5.01345446703866898173053257845, 5.71541974557285538754161035881, 6.17552078642226912621702421708, 6.85784332567369613649223781990, 7.86954311299612431794326407575
