| L(s) = 1 | + 1.44·2-s + 3-s + 0.0978·4-s − 0.404·5-s + 1.44·6-s + 2.87·7-s − 2.75·8-s + 9-s − 0.585·10-s − 11-s + 0.0978·12-s + 4.17·14-s − 0.404·15-s − 4.18·16-s + 1.23·17-s + 1.44·18-s + 4.83·19-s − 0.0395·20-s + 2.87·21-s − 1.44·22-s + 5.99·23-s − 2.75·24-s − 4.83·25-s + 27-s + 0.281·28-s + 4.74·29-s − 0.585·30-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.577·3-s + 0.0489·4-s − 0.180·5-s + 0.591·6-s + 1.08·7-s − 0.974·8-s + 0.333·9-s − 0.185·10-s − 0.301·11-s + 0.0282·12-s + 1.11·14-s − 0.104·15-s − 1.04·16-s + 0.299·17-s + 0.341·18-s + 1.10·19-s − 0.00885·20-s + 0.628·21-s − 0.308·22-s + 1.24·23-s − 0.562·24-s − 0.967·25-s + 0.192·27-s + 0.0532·28-s + 0.882·29-s − 0.106·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.192094061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.192094061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 5 | \( 1 + 0.404T + 5T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 - 5.99T + 23T^{2} \) |
| 29 | \( 1 - 4.74T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 + 6.08T + 37T^{2} \) |
| 41 | \( 1 + 2.29T + 41T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 + 1.93T + 47T^{2} \) |
| 53 | \( 1 + 4.21T + 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 2.88T + 71T^{2} \) |
| 73 | \( 1 + 6.31T + 73T^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 - 3.55T + 83T^{2} \) |
| 89 | \( 1 - 5.50T + 89T^{2} \) |
| 97 | \( 1 - 3.21T + 97T^{2} \) |
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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
−8.340814621622059404920554956389, −7.40627119910217724687154427374, −6.71833807027181578911792414855, −5.70700883948799970754786668848, −4.99549693292990888352335101598, −4.65932547102130359880028998062, −3.64247017076357303103184137299, −3.09408574672063467215629973245, −2.14535532965625629541071914345, −0.947801250026557588451601509508,
0.947801250026557588451601509508, 2.14535532965625629541071914345, 3.09408574672063467215629973245, 3.64247017076357303103184137299, 4.65932547102130359880028998062, 4.99549693292990888352335101598, 5.70700883948799970754786668848, 6.71833807027181578911792414855, 7.40627119910217724687154427374, 8.340814621622059404920554956389
