| L(s) = 1 | − 2.30·2-s − 1.91·3-s + 3.33·4-s + 3.27·5-s + 4.43·6-s − 1.87·7-s − 3.08·8-s + 0.683·9-s − 7.56·10-s − 2.21·11-s − 6.39·12-s − 3.64·13-s + 4.34·14-s − 6.28·15-s + 0.447·16-s + 3.59·17-s − 1.57·18-s + 1.47·19-s + 10.9·20-s + 3.60·21-s + 5.12·22-s + 7.78·23-s + 5.91·24-s + 5.71·25-s + 8.42·26-s + 4.44·27-s − 6.26·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 1.10·3-s + 1.66·4-s + 1.46·5-s + 1.80·6-s − 0.710·7-s − 1.08·8-s + 0.227·9-s − 2.39·10-s − 0.668·11-s − 1.84·12-s − 1.01·13-s + 1.16·14-s − 1.62·15-s + 0.111·16-s + 0.872·17-s − 0.372·18-s + 0.339·19-s + 2.44·20-s + 0.787·21-s + 1.09·22-s + 1.62·23-s + 1.20·24-s + 1.14·25-s + 1.65·26-s + 0.855·27-s − 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6325978241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6325978241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 + 3.64T + 13T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 - 1.47T + 19T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 - 4.86T + 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 7.72T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 59 | \( 1 - 3.58T + 59T^{2} \) |
| 61 | \( 1 + 0.904T + 61T^{2} \) |
| 67 | \( 1 + 8.44T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 3.01T + 73T^{2} \) |
| 79 | \( 1 + 2.84T + 79T^{2} \) |
| 83 | \( 1 + 5.76T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
| 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
−7.68216479984508107373502826616, −7.26169734344990994518720225781, −6.48495049490506440221928204860, −5.94737468023859238541167054524, −5.35173202655241776685123427329, −4.66286409601890617521223060101, −2.85611050294926366381716677450, −2.55169633703873119725307185917, −1.29394869337774809919790459068, −0.59519988697226478587405607450,
0.59519988697226478587405607450, 1.29394869337774809919790459068, 2.55169633703873119725307185917, 2.85611050294926366381716677450, 4.66286409601890617521223060101, 5.35173202655241776685123427329, 5.94737468023859238541167054524, 6.48495049490506440221928204860, 7.26169734344990994518720225781, 7.68216479984508107373502826616
