| L(s) = 1 | + 2-s − 3-s + 4-s − 3.77·5-s − 6-s + 2.13·7-s + 8-s + 9-s − 3.77·10-s + 4.21·11-s − 12-s − 13-s + 2.13·14-s + 3.77·15-s + 16-s + 4.66·17-s + 18-s + 8.26·19-s − 3.77·20-s − 2.13·21-s + 4.21·22-s + 0.374·23-s − 24-s + 9.28·25-s − 26-s − 27-s + 2.13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.69·5-s − 0.408·6-s + 0.808·7-s + 0.353·8-s + 0.333·9-s − 1.19·10-s + 1.27·11-s − 0.288·12-s − 0.277·13-s + 0.571·14-s + 0.975·15-s + 0.250·16-s + 1.13·17-s + 0.235·18-s + 1.89·19-s − 0.845·20-s − 0.466·21-s + 0.898·22-s + 0.0780·23-s − 0.204·24-s + 1.85·25-s − 0.196·26-s − 0.192·27-s + 0.404·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.604338119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.604338119\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 - 4.21T + 11T^{2} \) |
| 17 | \( 1 - 4.66T + 17T^{2} \) |
| 19 | \( 1 - 8.26T + 19T^{2} \) |
| 23 | \( 1 - 0.374T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 3.35T + 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 7.02T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 + 3.53T + 53T^{2} \) |
| 59 | \( 1 - 5.68T + 59T^{2} \) |
| 61 | \( 1 + 5.36T + 61T^{2} \) |
| 67 | \( 1 - 8.01T + 67T^{2} \) |
| 71 | \( 1 + 8.03T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
| 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
−7.85145673535321072758950329929, −6.92974912073963712783861672431, −6.70846269603447559793130432314, −5.41124824184289995761854490384, −5.03533171958925008689647776637, −4.33766022410683155609294213047, −3.54533319481127734410869829649, −3.17020038179459397737529317561, −1.55161899175472503763836179296, −0.813625304045214819260809637320,
0.813625304045214819260809637320, 1.55161899175472503763836179296, 3.17020038179459397737529317561, 3.54533319481127734410869829649, 4.33766022410683155609294213047, 5.03533171958925008689647776637, 5.41124824184289995761854490384, 6.70846269603447559793130432314, 6.92974912073963712783861672431, 7.85145673535321072758950329929
