| L(s) = 1 | + 2-s − 3-s + 4-s − 2.80·5-s − 6-s + 8-s + 9-s − 2.80·10-s − 11-s − 12-s + 2.80·15-s + 16-s + 7.96·17-s + 18-s − 6.48·19-s − 2.80·20-s − 22-s − 5.28·23-s − 24-s + 2.87·25-s − 27-s + 5.12·29-s + 2.80·30-s − 8.96·31-s + 32-s + 33-s + 7.96·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.25·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.887·10-s − 0.301·11-s − 0.288·12-s + 0.724·15-s + 0.250·16-s + 1.93·17-s + 0.235·18-s − 1.48·19-s − 0.627·20-s − 0.213·22-s − 1.10·23-s − 0.204·24-s + 0.574·25-s − 0.192·27-s + 0.952·29-s + 0.512·30-s − 1.61·31-s + 0.176·32-s + 0.174·33-s + 1.36·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637133907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637133907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2.80T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.96T + 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 + 5.28T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 5.09T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 0.322T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 0.871T + 59T^{2} \) |
| 61 | \( 1 - 2.15T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 + 7.44T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 1.61T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
| 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
−8.322893067985681849778390099941, −7.77355276847492292262559649407, −7.22879310715581657192938050079, −6.18265288791516062719991365611, −5.66638775860792254168616195747, −4.68664464047049222632433931200, −4.02297614110335763700332428568, −3.38250537199491794706764140625, −2.17400576853987307769344817120, −0.70023435125508330428256770482,
0.70023435125508330428256770482, 2.17400576853987307769344817120, 3.38250537199491794706764140625, 4.02297614110335763700332428568, 4.68664464047049222632433931200, 5.66638775860792254168616195747, 6.18265288791516062719991365611, 7.22879310715581657192938050079, 7.77355276847492292262559649407, 8.322893067985681849778390099941
