| L(s) = 1 | + 3.50·2-s + 4.26·4-s + 11.8·5-s + 22.4·7-s − 13.0·8-s + 41.3·10-s + 54.9·11-s − 62.3·13-s + 78.7·14-s − 79.9·16-s − 10.7·17-s − 33.3·19-s + 50.4·20-s + 192.·22-s − 25.0·23-s + 14.6·25-s − 218.·26-s + 95.9·28-s + 235.·29-s + 338.·31-s − 175.·32-s − 37.5·34-s + 265.·35-s + 223.·37-s − 116.·38-s − 154.·40-s + 399.·41-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.533·4-s + 1.05·5-s + 1.21·7-s − 0.577·8-s + 1.30·10-s + 1.50·11-s − 1.33·13-s + 1.50·14-s − 1.24·16-s − 0.152·17-s − 0.402·19-s + 0.563·20-s + 1.86·22-s − 0.226·23-s + 0.117·25-s − 1.64·26-s + 0.647·28-s + 1.50·29-s + 1.96·31-s − 0.968·32-s − 0.189·34-s + 1.28·35-s + 0.991·37-s − 0.498·38-s − 0.610·40-s + 1.52·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.359001585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.359001585\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 3.50T + 8T^{2} \) |
| 5 | \( 1 - 11.8T + 125T^{2} \) |
| 7 | \( 1 - 22.4T + 343T^{2} \) |
| 11 | \( 1 - 54.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 338.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 399.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 180.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 301.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 257.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 192.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 825.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 258.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 405.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 762.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 332.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 241.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 990.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.69e3T + 9.12e5T^{2} \) |
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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
−8.759343888552483875130930262803, −7.933615617249267391096781094278, −6.73953998543014615757622311776, −6.24382974425799584491692265020, −5.42479787886527131979008756183, −4.52062868240826693063528760516, −4.30781306043155034509287476198, −2.81232004400537895125291406509, −2.12090808191363266393047064293, −0.996416944651744282598350247635,
0.996416944651744282598350247635, 2.12090808191363266393047064293, 2.81232004400537895125291406509, 4.30781306043155034509287476198, 4.52062868240826693063528760516, 5.42479787886527131979008756183, 6.24382974425799584491692265020, 6.73953998543014615757622311776, 7.933615617249267391096781094278, 8.759343888552483875130930262803
