| L(s) = 1 | + 2-s − 3-s + 4-s + 0.0978·5-s − 6-s + 2.61·7-s + 8-s + 9-s + 0.0978·10-s + 6.41·11-s − 12-s − 1.11·13-s + 2.61·14-s − 0.0978·15-s + 16-s + 4.91·17-s + 18-s + 0.0978·20-s − 2.61·21-s + 6.41·22-s − 1.58·23-s − 24-s − 4.99·25-s − 1.11·26-s − 27-s + 2.61·28-s − 4.65·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.0437·5-s − 0.408·6-s + 0.989·7-s + 0.353·8-s + 0.333·9-s + 0.0309·10-s + 1.93·11-s − 0.288·12-s − 0.309·13-s + 0.699·14-s − 0.0252·15-s + 0.250·16-s + 1.19·17-s + 0.235·18-s + 0.0218·20-s − 0.571·21-s + 1.36·22-s − 0.330·23-s − 0.204·24-s − 0.998·25-s − 0.218·26-s − 0.192·27-s + 0.494·28-s − 0.865·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.996731693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.996731693\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 0.0978T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 6.41T + 11T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 17 | \( 1 - 4.91T + 17T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 + 7.92T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 - 6.50T + 41T^{2} \) |
| 43 | \( 1 + 4.13T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 - 4.65T + 53T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 1.75T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 6.35T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−9.277659434747393978254854787953, −8.034291297820137695312118151924, −7.45665405228975476917177877439, −6.53740019725372545902443308378, −5.81633484135487219805771027758, −5.14389161109692244171004619837, −4.14607914051482094517267216392, −3.64911110754828034275158084033, −2.04964327780205922654870096549, −1.18204822622538835051008343010,
1.18204822622538835051008343010, 2.04964327780205922654870096549, 3.64911110754828034275158084033, 4.14607914051482094517267216392, 5.14389161109692244171004619837, 5.81633484135487219805771027758, 6.53740019725372545902443308378, 7.45665405228975476917177877439, 8.034291297820137695312118151924, 9.277659434747393978254854787953
